Tracker is sometimes used as is in French, and often pronounced like traqueur.
If you want something more French, suivi would be close:
Logiciel de suivi des vaccinations contre le covid-19.
Site de suivi des vaccinations...
The government site about it is precisely using Suivi de la vaccination (3rd tab).
Suivi is more like "tracking" but unfortunately, suiveur (closer to "tracker") wouldn't work well here.
Note that French has a verb semantically closer to the verb "to track", pister but only used when the location of something is tracked. Pister means suivre à la trace (from piste = track).
Pister les vaccinations wouldn't make much sense but pister les déplacements d'un super-contaminateur (track a super-spreader) or pister un animal (track an animal) would. There is also dépister that means find the evidence of the presence of something (to screen): Dépister le covid-19.
In a broader way, the Merriam-Webster lists the following meanings for the verb "to track":
1 a: to follow the tracks or traces of : TRAIL
b: to search for by following evidence until found track down the source
2 a: to follow by vestiges : TRACE
b: to observe or plot the moving path of (something, such as a spacecraft or missile) often instrumentally
3 a: to carry (mud or other material) on the feet and deposit
tracking mud into the kitchen
b: to make tracks upon
4: to keep track of (something, such as a trend) : FOLLOW
5: to travel over : TRAVERSE track a desert
Here are French verbs close to each acceptation:
1 pister, suivre à la trace, prendre en filature
2 suivre, (tracer)
3 laisser des traces (de son passage)
None of them really match faire l'état des lieux which, being more static, doesn't convey well the evolving/moving meaning of track.
Finally, both Google translate and DeepL (the best online translation service so far) currently translate "Covid-19 vaccination tracker" to Suivi de la vaccination Covid-19 (DeepL) (Google). Reverso gives Outil de suivi de la vaccination contre la Covid-19 which is a little verbose but not bad anyway as it properly carry the meaning of tool while suivi is not the tool but what the tool achieves.