As some of us aren’t fluent enough in English to get the possible nuances, you might want to explain what is exactly meant by:

I care about socially vulnerable groups.

Here are suggestions to translate it, depending on the expected meaning:

Je me fais du souci à propos de groupes de personnes socialement vulnérables.

Je m’occupe de personnes socialement vulnérables.

Je m’intéresse aux groupes de personnes socialement vulnérables.

For you second question, the best option would be :

Je prends soin de mes clients.

Je pense à… doesn’t necessarily convey you care about them, that might just mean you don’t forget about them.

Je me préoccupe de… usually means there is an ongoing problem which is not necessarily the case with "I care".

You can use “je me soucie de” in both cases :

Je me soucie des groupes socialement vulnérables.

Je me soucie de mes clients.

As mentioned by jilliagre, “Je pense à” and “Je me préocuppe” don’t have the same meaning.

Je me préoccupe des groupes sociaux vulnérables would be the closest translation. However we have no equivalent in french for the exact meaning of care, which is neither se préoccuper (who can be understood as se faire du souci) nor s’intéresser (who can be understood as vouloir étudier, apprendre sur le sujet) but somewhere between those words. I’m interested, preoccupied, I feel empathy.

For customers I would say je prends soin de mes clients or Je fais attention à mes clients. But once again those words don’t perfectly render the meaning of care.

I believe that “Je m’engage auprès des groupes…” ou “Je soutiens les groupes…” reflects better what “I care” really means in this context. There is no exact equivalent in French.

I agree with the other answers that note that the context of your first, “socially vulnerable groups,” example is not totally clear, especially since you preface that example with the fact that you volunteer [to help such groups?], which notion of “currently volunteering” might lead one to think that by “to care about” you actually mean “to care for” or “to take care of.”

However, because you expressly use “to care about” (and not “to care for” or “to take care of”) in your actual example, I’m going to go out on a limb and interpret your example as giving a “reason” (i.e., that you care about them) for why you volunteer and not as a [simple] restatement of the fact that you do volunteer for/take care of/care for/ these groups in the first place.

With this admittedly assumptive interpretation (along with the arguably even greater assumption that you might actually “care deeply about” this issue and these groups) in mind, …

… I think you could use “prendre/avoir à cœur” to
express the notion of “someone caring [deeply] about” something” (as in the structure of your example where you are caring [deeply] about [the problems/situations facing] such groups), as follows:

“Je prends/J’ai les [problèmes/situations des] groupes
vulnérables à cœur.” …

… or I think you could also consider using “tenir à cœur” to express essentially the same idea, but in an inverted (nearly passive) form where “something (this issue) is affecting/has affected someone (you) to the point where [they/you] (now) care deeply about it,” as follows:

Les [problèmes/situations des] groupes vulnérables me tiennent à
coeur.

See:

[Avec une valeur intensive] Avoir/prendre (qqc.) à
cœur.= Y prendre beaucoup
d’intérêt.
(from CNRTL-TFLi)

and

Tenir à
cœur =
“Intéresser fort ; avoir beaucoup d’importance sur le plan personnel.”
(from fr.wiktionary)

Although I think the above uses of “à cœur” could work in your first example, they would probably be too much of an overstatement with your second, “restaurant clients” one, but maybe you could tone it back a bit with the more literal (albeit less spot-on) “prendre en [sérieuse] consideration” to use in that example:

Je prends [toujours] en [sérieuse] considération [les
besoins/attentes/désirs de] mes client/e/s.