More problems than gains
There exist sub-calibre bullets for firearms today, specifically for rifles, though these are usually made of Wolfram (Tungsten), not Depleted Uranium. The main issue though — that makes your concept moot — is that the body is fragile, and can only handle so much recoil before the poor squishy human — that fires the weapon — breaks. And you reach that limit far before you get any practical upshot of using Depleted Uranium (DU) bullets.
Dense sub-calibre munitions do exist, but not for added impact
A sub-calibre bullet of high density does not pack more of a punch. Instead, it allows the bullet to fly faster, which makes for a flatter trajectory and shorter time to target, which makes the target easier to hit. I have personally seen the Swedish military soldier handbook for sharp shooter rifles — such as the L96A1 AW — where the difference between using full caliber and sub-calibre ammunition is shown. Unfortunately I do not have it available too show you so you will have to take my word for it that the difference is quite marked.
The same thing goes — partially — for tank ammunition. Yes, it is true that the chemical / metallurgical qualities of DU are advantageous when it comes to defeating armor, as is the vastly improved penetrating depth when using dense materials. But the improved ballistics of sub-calibre ammunition is also of great importance in that it makes it easier for the gunner to hit at greater range.
Anyway, back to squishy humans. You are asking: can you use DU to increase the "punch" of a weapon so that weapons that can be lugged around by humans become a viable threat to armor?
Short answer: no.
Depleted Uranium is not a magic force multiplier. You still need to impart kinetic energy on the bullet. As this happens you will create recoil. And if you are trying to brute force your way though armor, you will create so much recoil that a human cannot handle it. This means you cannot increase the weapon's effect much by using DU.
Statically mounted heavy weapons already pack more "oomph" than your fictional soldier-carried weapons do. And then there are rocket propelled grenades, and anti-armor missiles, which do not rely on brute force but instead on clever explosives to make their way through the armor. Not to mention that missiles are guided and thus can be fired far outside the engagement envelope for your ballistic weapon. The thing is that modern armor is designed with these kinds of weapons in mind. And since these weapons pack more of a punch than that of your fictional firearm, it makes your weapon moot because it does not add anything of value.
Then — as a bonus — you have the problem of the metal being slightly radioactive which will create all sorts of "fun" problems. No, not acute radiation sickness, but let us just say that having the common soldiery run around on the battlefield with fissile material will make the world community raise an eyebrow or two. Not to mention that their travels home after the war will be "interesting" when they set off alarms left and right at airports, ports, and border crossings.
So, to summarize: can traditional firearms be made a threat to armour by the use of Depleted Uranium ammunition?
No, they can not. Mechanised forces are already expecting you to hit them with much beefier ordnance than that, and are up-armoured for that eventuality, leaving your ballistic handhelds woefully inadequate to be a threat.