Using technology already available to them at the time, could the 14-15th century world create a somewhat effective tank/AFV that could either be used to move artillery quicker around the battlefield or provide cover fire for troops to charge?
Leonardo da Vinci designed what would be considered the prototype of all AFV's in the 1490's, and after fixing an error in the drawings for the gearing, modern reproductions have actually demonstrated the ability to move over firm, level ground and withstand the shock of firing the multitude of small cannon around the perimeter.
However, there are a multitude of issues which make this a non starter. The first issue is the power source, several crew members turning hand cranks geared to the wheels. It is unlikely they would have been able to generate enough power to cross broken terrain, climb hills or otherwise move anywhere except firm, hardpacked ground.
The second and related problem is the ground pressure would be extremely high given the narrow wooden wheels. The vehicle only sits on 4 wheels, so there would be a considerable mass concentrated on 4 very small contact patches. While it might be possible to build wider wheels, the issue then becomes the weight of the overall vehicle also increases, further reducing the power to weight ratio.
Leonardo, like all inventors prior to the Industrial revolution, was limited to the power of human or animal muscles. The compact size and shape of the AFV would have prevented placing horses inside, and if a team was hitched to the vehicle like a large wagon, the enemy would quickly realize that killing the horses would immobilize the vehicle.
Steam engines were still a long way off. The first modern steam engine was designed and demonstrated by https://en.wikipedia.org/wiki/Jerónimo_de_Ayanz_y_Beaumont (1553 – 23 March 1613) in 1606. Newcomen and Watt steam engines were more powerful and practical, but not introduced until the 1700's.
There were "engines" dating as far back as the first century AD, but these simple atmospheric engines described by Hero of Alexandria were considered toys, and the most practical application seemed to be automatic door openers to thrill people as they arrived at temples. The amount of power these or Newcomen steam engines could produce was rather limited, and the extra mass of the engine, boiler, fuel and "black gang" to run them likely would have completely overwhelmed any possible advantage they could provide. Indeed the problem was so intractable that steam powered "traction engines" for farms were very slow and unwieldy, and no steam powered AFV was ever introduced into service, even though such a thing might have been theoretically possible in the American Civil War.
So there is no practical way to create a tank of armoured fighting vehicle in the 1400-1500 time period.
Not only they could: they actually did
It happened that at the beginning of the 15th century, a Bohemian (that would be a Czech in modern terms) priest and theologian named Jan Hus "denounced what he judged as the corruption of the Church and the Papacy" (Wikipedia), and promoted some mild reformist ideas. The Catholic Church being the Catholic Church could not help itself, and, while Hus was gathering popular support, decided to raise some money for a totally unrelated war by selling indulgences in Bohemia. This did nothing but add fuel to the popular support for Hus.
So what could the poor oppressed Catholic Church do? They had the seditious preacher arrested, and burned him at the stake.
Which generated a general uprising in Bohemia. The fighting soon took the shape of a full-scale war, lasting for 25 years, from 1419 to 1434; the Hussites "defeated five consecutive crusades proclaimed against them by the Pope (1420, 1421, 1422, 1427, 1431), and intervened in the wars of neighboring countries" (Wikipedia).
In military history, the Hussite Wars are notable for at least two reasons:
They were the first wars featuring "extensive use of early hand-held firearms" (Wikipedia); and
They were the first wars featuring extensive use of war wagons.
A medieval European war wagon was developed by the Hussite Wars around 1420 by Hussite forces rebelling in Bohemia. It was a heavy wagon given protective sides with firing slits and heavy firepower from either a cannon or a force of hand-gunners, archers and crossbowmen, supported by infantry using spears, pikes and flails. Groups of them could form defensive works, but they also were used as hardpoints for formations or as firepower in pincer movements.
This early use of gunpowder and innovative tactics helped a largely peasant infantry stave off attacks by the Holy Roman Empire larger forces of mounted knights. After the Hussite wars, they stayed in usage as the special tactical weapon of Bohemian mercenaries which prevailed against knights or foot soldiers.
Its successful history came to an end, at least for large scale engagements, with the development of field-piece artillery: a battle wagon wall "fortress" of approximately 300 wagons was broken at the Battle of Wenzenbach September 12, 1504 by the culverines (early cannons) and muskets of the landsknecht regiment of G. Frundsberg.
(Wikipedia, s.v. War wagon)
Not content with the introduction of the first armored combat vehicles, the Hussites went and developed specialized tactics for the use of their new war machines:
In the 15th century, during the Hussite Wars, the Hussites developed tactics of using the tabors, called vozová hradba in Czech or Wagenburg (wagon town) by the Germans, as mobile fortifications. When the Hussite army faced a numerically superior opponent, the Bohemians usually formed a square of the armed wagons, joined them with iron chains, and defended the resulting fortification against charges of the enemy. Such a camp was easy to establish and practically invulnerable to enemy cavalry.
The crew of each wagon consisted of 18 to 21 soldiers: 4 to 8 crossbowmen, 2 handgunners, 6 to 8 soldiers equipped with pikes or flails, 2 shield carriers and 2 drivers. The wagons would normally form a square, and inside the square would usually be the cavalry. There were two principal stages of the battle using the wagon fort: defensive and counterattack. The defensive part would be a pounding of the enemy with artillery. The Hussite artillery was a primitive form of a howitzer, called in Czech a houfnice, from which the English word howitzer comes. Also, they called their guns with the Czech word píšťala (hand cannon), meaning that they were shaped like a pipe or a fife, from which the English word pistol is possibly derived. When the enemy would come close to the wagon fort, crossbowmen and hand-gunners would come from inside the wagons and inflict more casualties on the enemy at close range.
(Wikipedia, s.v. Wagon fort)
Isn't history wonderful?
It would not move fast enough to be effective.
Tanks must be balanced within the "triangle" of firepower, armor, and mobility. Firepower and armor must be seen in proportion to the rest of the battlefield, but pre-industrial technology gives not enough mobility. Consider the siege towers or protective roofs over battering rams. They could only be used against stationary targets like castle walls. Also consider the Hussite war wagons of your era.
I realise you’re probably asking about some steampunk style tank, but in terms of usage and classification, I’d argue they did have them:
Heavy Cavalry was effectively the AFV of the 14-15th century.
Because of the armour on the horse, I’d argue it’s a proper AFV, not just a light vehicle like a ‘technical’ (hilux + GPMG - the horseback archer being the ancient equivalent). The armour on heavy cavalry made them relatively immune to arrows, crossbows etc. Though not to cannon fire.
However, I’m not aware of any attempts to use heavy cavalry as firing points for cannons. I think the issues may have been recoil, and effectiveness. A recoilless cannon should be lighter and easily fired from horseback; I’m unsure whether such devices were creatively with 14-15C technology though.
Why were they not used in a similar role as tanks? Mainly I think because of their mobility which meant they excelled in breaking infantry, and the lack of an effective weapon which would justify its deployment on heavy cavalry (as opposed to horse archers).
Nothing I’m aware of would have stopped heavy cavalry from pulling artillery, but due to the range of artillery and the inaccuracy of long-range weapons, there was no need for armoured horses in this role.
Heavy cavalry also played a similar role to tanks in providing shock to the enemy and a morale boost to allied troops; however the strong class divide between knights and expendable infantry may have negated this somewhat...
War Elephants would be another example of an early AFV. The psychological effect would be much stronger compared to a Knight, similar to a MBT, and the slower pace of an elephant would mean it played more of a supporting role. Examples abound of shooting from the castle, including (at least later) fairly large bore weapons.
I’m not sure how heavily armoured the elephant was (the castle could be armoured), but elephants are pretty thick skinned - immune to muskets, but still vulnerable to cannons, though this wasn’t an issue until the late 15th century.
The poor availability of elephants in Europe limited their use, but where they’re available, they were used widely.