"I just need Sam's approval" is correct, the other sentence is not, for several reasons. You can forget about "approbation". While the word exists, it is only used in formal language, and even then, it's infrequent. Just say "approval". Feel free to ask follow-up questions.

I've never used approbation either online or in speech, so chances are there aren't any good generic cases for using it. Approval is pretty generic, usually used in a contractual or businesslike context. Another phrase that often means something similar is to "sign off" Ex. "I just need Sam to sign off on those changes." if instead you are using approval to mean "acceptance" or "praise" then those words and other synonyms are preferred.

I just learned a new word approbation. I think this word is hardly used.

I agree with the people above that ‘approbation’ is rarely used.

The best sentence would be the first one "I just need Sam's approval." If you wanted to use approbation you would basically make the same sentence but switch the words, "I just need Sam's approbation." or you could say, "I just need the approbation of Sam." "by Sam" doesn't work. The biggest difference between approval and approbation is honestly just commonality. You're very unlikely to hear someone say approbation in casual conversation or even in business because it is very formal and pretty uncommon. Grammatically approbation is a little stronger than approval and is more about the act of approving. Approval is more about giving permission or acknowledging someone or something. They are synonyms and are interchangeable but I would recommend using approval rather than approbation because approval is easier to fit into a regular sentence and it is way more common especially in spoken American English. Hope this helps!