A verb for "picking small bits of food from the tray or the main plate before it is served or others have started eating"

The first word that occurs to me is sample:

• a small part or quantity intended to show what the whole is like.

• try the qualities of (food or drink) by tasting it.

  • synonyms: try (out), taste, test, put to the test, experiment with; appraise, evaluate, test drive, check out

  • "we sampled the culinary offerings"

• get a representative experience of.

Kids! Do not sample the food.

Don't sample the food with your dirty finger.

He was sampling the chicken nuggets before everyone came to the table.

My parents (both of whom were US-born with roots in the Midwest) would use snitch for exactly this purpose:

To steal (something, usually something of little value); pilfer: snitched a cookie from the counter. (Free Dictionary)

However, the other usage of the word (to inform or tattle on someone) is probably much more common these days, so using "snitch" in this way may cause confusion in the reader/listener.

Other possible alternatives are pilfer and filch, though in my mind both of these have more of a sense of criminality than "snitch" does.

I don't believe a specific word exists but what comes to mind is filch See the Merriam Webster example

too hungry to wait until the party had started, he filched a cookie from the buffet table when no one was looking

It fits your examples 1) and 3) but I'm less comfortable with 2). There I believe grab or grab at would be more appropriate since the emphasis is not on taking sneakily but on the hand contact.

Consider. sneak

To move, give, take, or put in a quiet, stealthy manner: sneak candy into one's mouth; sneaked a look at the grade sheet.

American Heritage Dictionary

Billy snuck a cookie from the dessert table Purdue OWL

In your examples,

Kids! Do not sneak samples of the food. Go to the dining room and wait at the table.

Don't sneak scraps of the food with your dirty fingers. Go wash your hands.

He was sneaking bits of the chicken nuggets before everyone came to the table.