To effectively regenerate, someone needs both massive amounts of calories, stamina or more generally energy (to sustain the accelerated metabolic process) and a sufficient quantity of matter to replace damaged bodyparts. Most movies and works of fiction apply a suspension of disbelief, which is quite sufficient for the average viewer (in the sense that they do not wish to question the "whys" and "hows" unless they become glaringly absurd).
Now, we need to couple this with the shape-shifter baggage. This law dictates shape-shifters should not be able to assume massively larger or smaller forms. For example, a Skrull should not be able to morph into a lifelike cat, given the difference in size (then again, Talos implies he could do similar extravagant feats... but he's also supposed to be a particularly talented and experienced Skrull). In other words, this law dictates shape-shifters cannot summon matter from nowhere or make it disappear just to morph more accurately.
Combined, these two laws outline that unless Skrulls can produce biological matter out of nowhere, they cannot actually regenerate. That said, there is nothing to prove or disprove that they can or cannot close cuts, heal up bruises or maybe reattach lost limbs. This does not really involve creating matter and thus could be doable, depending on their morphing's actual mechanism.
I should also add that, as far as I remember, every serious non-lethal injury inflicted on a Skrull was done through photon blasts or laser rifles ; as least, those we saw (this is discarding Yon-Rogg throwing heaps of debris at Skrull attackers, for example). These weapons instantly cauterize wounds, which traditionally is an effective way of disabling regeneration or shape-shifting.