The second one sounds fine. The meaning of the first one is unclear.

I sneaked into the kitchen and ate his snacks.

You can say: I snuck some of his snacks. I sneakily took his snacks. ‘Secretly’ is ok, but it’s a little grandiose for something as simple as taking someone’s snacks. It’s not necessary or natural to use ‘eating’ here because the important fact is that you took them. (‘Eating’ is understood without saying it) In my first sentence, ‘snuck’ would be understood to mean ‘took sneakily’

Both are correct but the first is less natural. In #1, "eating his snacks" is a noun clause that is the direct object of the verb "sneaked". This is correct usage. One of the recognized meanings of the verb "to sneak" is "to convey in a stealthy way". For example, you can say "I sneaked a look at the performance" (in which "a look" is the thing conveyed) "I sneaked dancing the cha-cha into my schedule for the evening". (in which "dancing the cha-cha" is the thing that is conveyed into my schedule. So #1 is a correct and meaningful sentence, but many people will have trouble understanding it. One problem with the sentence is that it puts the most essential fact into a noun, "eating". It is better to put the force of your sentence into a verb than a noun: "I sneakily ate his snacks".

The second one is correct and natural. The first one is definitely a bit unclear and not a natural way of expression.