Maybe?

According to the fourth elementary rule of usage from Strunk and White's The Elements of Style, and section 6.28 of The Chicago Manual of Style, you are supposed to place a comma between two independent clauses connected by a conjunction, just like you said.

For example

• I like apples. (independent clause)
• He prefers oranges. (independent clause)
• I like apples, but he prefers oranges. (two independent clauses separated by a comma and the coordinating conjunction "but")

CMOS states that this rule still stands if the subject is implied, again just as you've said.

For example

• Go to the store. (independent clause with implied subject of "you")
• Buy some cheese. (independent clause with implied subject of "you")
• Go to the store, and buy some cheese. (two independent clauses separated by a comma and the coordinating conjunction "and")

However, because the subject of both sentences is the same, you could also read it as

• Go to the store and buy some cheese. (compound predicate with implied subject of "you")

In which case section 6.29 of CMOS states that no comma is needed, except to prevent ambiguity in a case like

• She recognized the man who entered the room, and gasped. (comma included despite compound predicate to indicate that "she," not "the man," gasped)

Our original example has the added complication of being a question, but I my first thought would be to read it as a compound predicate so that the whole bit describes the "what." So it would be

• What can you [steal and not get in trouble]? (with the verbs "steal" and "get in trouble" being compounded by "and")

You could also read it as a question of whether the bit following the comma is restrictive (essential to the meaning of the sentence, no comma allowed) or non-restrictive (merely supplementary information, comma needed) [CMOS section 6.31], in which case I would say that the bit following the comma in our example is absolutely essential to the riddle, so no comma should be included.

HOWEVER, if we get back to basics section 6.16 of CMOS introduces commas by stating that "effective use of the comma involves good judgment, with ease of reading the end in view." I seriously doubt that the author spent any time at all considering implied subjects or restrictive clauses. Instead I read their use of the comma as most likely intended to represent a dramatic pause in a clever riddle.

So, punctuation is pretty subjective.

Most grammar "rules" are just guidelines to help people try to communicate more clearly. I've cited CMOS here, but that is just a STYLE guide, and you could make up your own if you wanted to. In the field of linguistics grammar is basically "whatever sounds okay to a native speaker."

I'm not an authority on any of this, but as you can probably tell from this response I AM the kind of freak who has lost sleep over the "should you put a comma between independent clauses with implied subjects that are connected by coordinating conjunctions?" conundrum, so thanks for this opportunity organize my thoughts.