What a Wild-Shaped druid can physically do is up to the GM
The rules on Wild Shape include the following (emphasis mine):
- (…) You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can’t use any of your special senses, such as darkvision, unless your new form also has that sense. (…)
While that doesn’t really say anything about this particular case, it does show us that the physical capabilities of a Wild-Shape form do matter. And with that, we can also see that skill checks (like most things) require the character to physically do something and whether a given Wild-Shape form can physically perform that same thing is up to the GM.
Regarding Wisdom checks and Medicine checks in particular, we know very very little:
A Wisdom check might reflect an effort to read body language, understand someone’s feelings, notice things about the environment, or care for an injured person. The Animal Handling, Insight, Medicine, Perception, and Survival skills reflect aptitude in certain kinds of Wisdom checks.
A Wisdom (Medicine) check lets you try to stabilize a dying companion or diagnose an illness.
And yeah… that’s all we know. Ultimately, the character does not say “I perform a medicine check” they say “I attempt to stabilize my friend” and the GM can ask (perhaps even should ask) how they plan to do that and from there they can make a ruling about the feasibility of that action and determine its DC based on the likelihood of its success.
For me, this depends almost entirely upon the druid’s form, because even if you know how to stabilize somebody, doing so while the size of a spider or the weight of an elephant simply will not work (well, I’ve had my players surprise me more times than I can count, so maybe they really will come up with a way for a spider to stabilize their friend). There would thus be times where I would really need to be convinced that it could work and it’d be very likely that no roll would be made and there would also be times where I would say it might work and would call for a roll.