Learn Swedish: Jag känner till problemet. - I am aware of the problem.

If you just use känner, it usually means you "feel" something or "know" a person. To say you are "familiar with" or "know about" a situation, thing, or concept, you need the phrasal verb känner till. Saying Jag känner problemet would sound like you are personally acquainted with the problem as if it were a human!

Could I use vet instead of känner till in this sentence?

Not quite. Vet is used for knowing hard facts, clauses, or specific information (like knowing what time the train leaves). Känner till means you are aware of something's existence or familiar with it. You are saying you are aware of the problem, so känner till is the natural choice.

Where does the stress go when pronouncing känner till?

Because känner till is a Swedish phrasal verb, the emphasis falls on the particle, which is till. You should stress till rather than känner when speaking this sentence.

Why does problem take the -et ending here?

Problem is an ett-word (ett problem). Because it ends in a consonant, you add -et to the end to make it definite, creating problemet, which means "the problem".