Learn Finnish: Hallusinaatio saa hänet seisomaan transsissa. - The hallucination gets him to stand in a trance.

This is the core structure for causation. When you use saada to express getting or making someone do something, it strictly requires the 3rd infinitive in the illative case (-maan/-mään). If you used the basic infinitive like in hän saa seisoa, the meaning would change entirely to he is allowed to stand.

Why is the object hän written as hänet here?

Personal pronouns like hän have a specific accusative form ending in -t when they act as the total object of a sentence. Because the action of getting him to stand is a fully completed, successful action, we use the total object hänet.

Is transsissa a normal way to express being in a state of mind?

Yes. The inessive case (-ssa/-ssä) translates to in and is frequently used in Finnish to express being in a physical or mental state. Just like you can be in a house, you can be in a trance (transsissa), in a panic (paniikissa), or in a coma (koomassa).

Could saada hänet seisomaan also be translated as allowing him to stand?

No, the meaning is strictly causative here. It only means getting or making him stand. To express allowing or letting him stand, you would need a different verb structure, typically using the verb antaa (to let/give): antaa hänen seisoa.