Learn Finnish: Verokortti on tärkeä asiakirja. - The tax card is an important document.

You would conjugate the epistemic modal verb taitaa (to probably be/do) and change the main verb on back to its basic dictionary form (1st infinitive), which is olla. The sentence becomes: Verokortti taitaa olla tärkeä asiakirja.

Could I use mahtaa instead of taitaa to express that probability?

You could, but it changes the feeling of the sentence. While taitaa expresses a fairly confident probability ("is probably"), mahtaa expresses more of a guessing wonder. So, Verokortti mahtaa olla tärkeä asiakirja translates better to "I wonder if the tax card is an important document."

Why are tärkeä and asiakirja in the basic nominative case here instead of the partitive?

This is a standard "A is B" sentence. Because the subject (verokortti) is a singular, countable thing, the words describing it (the predicative) stay in the nominative case. If the subject were an uncountable mass noun like water or coffee, you would need the partitive (for example: Kahvi on tärkeää).

The word asiakirja looks like a compound word. What are its parts?

You are right! It is formed from asia (matter, thing, issue) and kirja (book). In Finnish, an "issue-book" or "matter-book" is the literal way to say "document".