Learn Finnish: Lääkäri kehitti ensimmäisen rokotteen vuonna 1796. - The doctor developed the first vaccine in the year 1796.

This is a specific rule for reading calendar years. When they immediately follow the essive word vuonna (in the year), the numbers themselves are spoken in the basic, uninflected nominative case (tuhat seitsemänsataa yhdeksänkymmentä kuusi). This is a welcome exception that saves you from inflecting a massive compound number!

Why is the object ensimmäisen rokotteen in the accusative (genitive-like) form rather than the partitive?

The verb kehittää (to develop) can take either a partitive or a total object depending on the context. Because the development was successfully finished (the first vaccine was fully created), the action is complete. Therefore, the object takes the total object -n ending, and the adjective ensimmäinen must agree with it.

The word for vaccine is rokote. Why does the stem change so much before adding the -n ending?

Nouns ending in -e have a special inflection pattern. When you add most grammatical endings to them, you double the final vowel and apply the strong grade of consonant gradation. So, the base form rokote gets an extra -e- and the single -t- strengthens to -tt-, creating the stem rokottee- before adding the -n accusative ending.

Could I use the inessive form vuodessa instead of vuonna to mean "in the year"?