Learn Russian: Я нигде не могу найти свой номерок!

Yes. Just like ни один (not a single) requires the negative particle не before the verb, negative adverbs like нигде (nowhere/anywhere) also require it. Russian uses strict double negation here, so you must say нигде не могу (literally 'nowhere I cannot').

Why use the reflexive pronoun свой here instead of мой (my)?

When the owner of the object is also the subject of the sentence (here, Я), Russian strongly prefers the reflexive possessive pronoun свой (one's own). While мой is grammatically possible, свой sounds much more natural to a native speaker in this context.

Why is the perfective verb найти used after могу instead of the imperfective?

The verb мочь (to be able to) can take either aspect depending on your intent. Because you are focused on the successful result of a single, specific action (finding the tag right now), the perfective найти is required. If you were talking about a recurring inability to find things in general, you would use the imperfective находить.

What exactly is a номерок? Is it just a small number?

While it literally translates to 'little number' (a diminutive of номер), in everyday Russian номерок specifically refers to a physical token, like a coat check tag, a cloakroom ticket, or a queue number. It is in the accusative case here as the direct object, but because it is an inanimate masculine noun, its form is identical to the nominative case.