As noted in comments, depth measurement is necessary but not sufficient for making charts. You also need to know where you are horizontally to know where on the chart to record that depth mark! That's the hard part; unfortunately most of this answer covers depth-finding ideas.
However, flying high in the air could give sight of land. That's possible with D&D 5e spells or for an 8th-level druid (Wild Shape into a flying animal).
Are there any likely ways of doing it with the sort of magic you get from access to low-level D&D spells?
D&D doesn't have a lot of spells for mapping; the main navigation spell is 6th lvl Find the Path which shows you how to get somewhere but not your own absolute location. There's also teleport to a known location. There are a few class features / skills for navigation on land (e.g. Ranger class stuff), but there are several spells for going underwater / talking to animals.
Perhaps your best option would be a druid that can Wild Shape into an animal with echolocation, like a dolphin. If this allows accurate depth mapping like sonar tech, you're all set. With other magic, people on the ship can communicate with them in dolphin form to relay and record measurements. (e.g. the Message cantrip cast by someone on the ship, which allows a telepathic reply. Or maybe a Speak With Animals spell.) Or give the dolphin a board of numbers to point at.
If we look at actual 5th edition D&D spells:
First level spellcaster character get first-level spells, 3rd level character for 2nd level spells,
2*n - 1 char level gets at least 1 spell slot of level
n. (See table for druids for example). And Cantrips are easy spells that can be cast an unlimited number of times per day, unlike higher level spells. But "most people" in a D&D setting aren't lvl1 of any class, and people of higher than 1st or 2nd level are rarer still.
The official sourcebooks don't have a depth-measuring cantrip, but it would reasonable for a wizard to research a new cantrip that just measures the depth of water with some limit like 300 feet. Some damage cantrips like Firebolt or Eldritch Blast have ranges of 120 ft, also Message. (You could maybe use such cantrips as range finders, to see if they reach the bottom...) A 1 mile limit would probably warrant being at least a 1st level spell, but could be something you could maintain concentration on for 10 minutes or an hour instead of just taking one depth sample in a single place.
The lowest level D&D spells with more than 120 foot range are Earthbind (2nd; 300 ft), Skywrite (2nd, "Sight", i.e. any point in the sky you can see), and Clairvoyance (3rd level, 1 mile). So it's rare for magic to affect or detect anything outside a pretty local area near you, until you get to higher level spells. But Clairvoyance actually lets you see or hear from that point for 10 minutes; a more specialized spell could trade more limitations for that range at a lower spell level. Or you could get cheesy and decide that the range is how far you can be from the point on the surface of the water, and the effect is telling you the water depth at that point. So the range doesn't have to reach to the bottom; that would be the area of effect of the spell.
There are some "feats" like Magic Initiate that let someone learn a cantrip (and a 1st level spell) without being an official Wizard or other spellcasting class, so if you're modelling a world after D&D you could have sailors / oceanographers who know a depth cantrip but not other spells with a weaker version of this feat.
Official pre-existing D&D spells let you communicate with animals (including aquatic ones), or go underwater yourself. Any of these can give you at least a rough idea of depth even without measuring equipment, or could help you use measuring equipment. If magic is highly available, it would be plausible that it gets used for this instead of just weighted sounding ropes, although ropes of known length do work well.
Cantrip: Light (Bard, Cleric, Sorcerer, Wizard) - make the weight at the end of your rope light up like a lantern ("bright light" in a 20ft radius), making it easy to visually see for some depth below the water.
1st level Find Familiar (Wizard) - You gain the service of a familiar, a spirit that takes an animal form you choose: ... including fish (quipper) and sea horse. While your familiar is within 100 feet of you, you can communicate with it telepathically. So that gives you a precise 100 foot reference point to tell if water is shallower than 100 ft, or just a bit, or a lot, deeper. Or if you're not at water level, +- whatever. Additionally, as an action, you can see through your familiar’s eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has.
1st level Speak With Animals (Bard, Druid, Ranger) will let you ask fish or aquatic mammals about the depths, as suggested by @David Hambling . If necessary, Animal Friendship is also 1st level, and will "charm" an animal for 24h. (Also a Bard, Druid, or Ranger spell)
2nd level Alter Self (Sorcerer, Wizard) lasts 1 hour, and has an Aquatic Adaptation option: gills / webbing: underwater breathing and a swim speed = walking speed. Presumably also tolerance for cold water, although that's not mentioned because D&D 5e doesn't usually bother about that for the intrepid heros who will be casting these spells on themselves / each other.
3rd level Water Breathing (Druid, Ranger, Sorcerer, Wizard) lasts 24h on up to 10 creatures, and can be cast as a ritual (not using up a spell slot but takes an extra 10 minutes to cast, plus the normal 1 Action ~= 1 round = 6 seconds). Repeated ritual castings of that could get whole crews of people able to descend a rope and swim / walk around on the bottom surveying. It doesn't help you swim better; if you don't have mundane fins + wetsuit you'd want 4th level Freedom Of Movement (Bard, Cleric, Druid, Ranger), lasts 1 hour.
D&D also doesn't make a big deal out of cold water, but in many oceans you'd need some protection from the cold. (Protection from Energy (Cold) is a 3rd level spell, single target and requires concentration.)
4th level Control Water (Cleric, Druid, Wizard) would let you "part" the water and make a trench up to 100 ft deep (100 ft cube of no water for up to 10 minutes), giving you easy view of the bottom if it's shallower than that, or not much deeper. The precise depth limit of the spell could be a useful depth reference. But it takes a spell slot so you can only do it a few times per long rest (day), depending on character level. A warlock that gets this spell somehow could do it 2 times per short rest (1 hour break during a day), or more at higher levels.
5th level Commune With Nature (Druid, Ranger) can give you knowledge of "terrain and bodies of water"; arguably you could use it to map the ocean floor from a ship in a 3 mile radius. It costs a 5th level (or higher) spell slot so you can only do it a few times per day. (Once for a 9th level caster). This is not a low-level spell.
Starting at 4nd level, a Druid's Wild Shape allows turning into an animal with a swim speed (for "a number of hours equal to half your druid level"). So you could become a dolphin, sea otter, shark, or other animal that's comfortable in the local water temperature. There's a tradeoff between having hands vs. needing to breathe air (aquatic mammal) or not having a very high sustained swim speed (octopus).
You retain your mind so you could totally use this to check out the bottom, like maybe swimming along the bottom while holding one end of a rope and keeping it directly below a ship. People on the ship can count knots / marks on the rope to record depth without having to stop and reel the rope all the way in between depth soundings. IDK if this is much better than just normal depth soundings with a weighted rope.
(Having a Wizard polymorph someone would be somewhat similar, although that replaces the person's physical and mental stats with the creature's stats for the duration. But if you only had a wizard, not a druid or ranger, it might be an option.)
Other non-spell class abilities
D&D also has lots of class abilities and features other than spells. One of the most relevant might be the Ranger's Favored Terrain / Natural Explorer feature, which they get at 1st level. If Coast applies to open water near the coast, this could be very valuable. Open ocean with its lack of landmarks is harder to justify as working with this feature, though; immunity to becoming lost is way too good.
You are particularly familiar with one type of natural Environment and are adept at traveling and surviving in such regions. Choose one type of Favored terrain: Arctic, coast, Desert, Forest, Grassland, Mountain, swamp, or The Underdark. When you make an Intelligence or Wisdom check related to your Favored terrain, your Proficiency Bonus is doubled if you are using a skill that you're proficient in.
While traveling for an hour or more in your Favored terrain, you gain the following benefits:
I'm sure other classes have some other relevant features.