star wars - How many seats were there in the Senate of the Galactic Republic?

We don't know. Sources are conflicted as to how many there were.

The best source I can find for a canon answer is the Databank article on the Galactic Senate, which simply says that there were "hundreds of politicians". By visual inspection, there do appear to be hundreds (perhaps thousands) of repulsorpods (I count about 20 in the part of the lowest row that is visible):

Moving to a pseudo-canon source, the Episode III novelization mentions the Delegation of 2,000, which was a group of senators who opposed Palpatine (the Delegation was also shown in a deleted scene from Episode III). Since the Jedi allege Palpatine has control of the Senate, these 2,000 senators are likely in the minority -- meaning there are possibly 10,000 senators in total. Presumably, each of these senators had their own repulsorpod.

Some sources, including Wookieepedia and Wikipedia, claim that there were exactly 1,024 repulsorpods. Neither source points to a source with that number -- Wikipedia links to the Databank article mentioned above, but that Databank article does not include the exact number. The number is probably based on the number of sectors that made up the Republic at the time of the Ruusan Reformation (Legends). The Ruusan Reformation Wookieepedia article in turn appears to get its number of 1,024 sectors from the Legends "Sector" article, which cites the Legends Star Wars: Attack of the Clones Incredible Cross-Sections reference book.