@OP
Most ordinary buses go to the big bus station in Aranyaprathet on the Thailand side.
https://wikitravel.org/en/Aranyaprathet
This is still 6km away from the border and you have to take a tuk-tuk the extra distance. Apart from the extra expense, the driver might take you some of the scam visa offices on the way. (There is also a sorngtairw, if you can find it).
A better option is to take one of the (fewer) buses that go to a market called 'Rong Kluea' (in the Eastern suburb of Aranyaprathet -Khlong Leuk). This is only about 100 metres from the border. There are buses direct to Rong Kluea from Morchit (I used to have the bus number(s) written down, but I can't find it now).
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I guess another question I have is this:
Is it possible to book a bus online from Bangkok to Poipet?
If you are looking for the convenience of booking in advance, I guess you could book the international bus and leave it at the border, if you don't mind wasting a bit of money. Tell the driver you are getting off!!
Maybe a better option for online booking is this site - it will let you book online a ticket from Morchit to Rong Kluea.
https://12go.asia/en/travel/morchit/rong-klua-market
(Some buses listed only go to Aranyaprathet and some go right to the border at Rong Kluea (0330, 0430, 0700, 0800, 0900, 1200 etc)). Take a Rong Kluea one.
...Or you could just rock up at Morchit and ask for the public bus to Rong Kluea. It's probably the same bus as the online one, but you will save some money (I think it cost about 250B). If people look blank, say 'Talat Rong Kluea glai Aranyaprathet' (Rong Kluea Market, near to Aranyaprathet).
