Learn Polish: Oferuję ci swoje wsparcie.

The verb oferować (to offer) directly takes the Dative case to show who is receiving the offer. We use the Dative for the direct recipient of an action (like giving, telling, or offering). You would use dla ciebie when doing a favor or getting something for someone's benefit, like buying them a gift.

Could I use the longer pronoun form tobie instead of ci?

You could, but ci is much more natural here. We use the short form ci in the middle of a sentence when it is just regular, unstressed information. You would only use tobie if you wanted to strongly emphasize 'you' specifically (for example, 'I am offering my support to YOU, not to him').

Why does it say swoje instead of moje (my)?

Swoje is a reflexive possessive pronoun that means 'one's own'. In Polish, when the owner of the object is also the subject of the sentence (here, 'I' am the one offering, and it is 'my' support), it is standard to use a form of swój. Using moje wouldn't be completely wrong, but swoje sounds much more natural to native speakers.

What case is the word wsparcie in, and why didn't its ending change?

It is in the Accusative case because it is the direct object of the sentence (it is the thing being offered). Wsparcie is a neuter noun, and for neuter nouns, the Accusative form looks exactly the same as the basic Nominative dictionary form.