There shouldn't be any issue with a wheelchair on the train but you can't travel to the Giants Causeway itself by train,you would need to transfer to a bus.
As far as i know None of the coach tours to the Giant's Causeway can accept wheelchairs, unless the passenger can transfer out of the chair.You would need to get a private tour by accessible taxi from central station.
You could also stay on the rails to portrush and book a taxi from their but you will miss out on the coast and glens.
but this is easy you just take the 402 bus from Coleraine train and bus station (its the same place same building) to the giants causeway, you get the train to coleraine from Belfast Great Victoria street train will be going to Londonderry or Portrush, translink are good at getting wheel chairs on and off trains and buses (translink are the public transport operator in Northern Ireland) the conductor will have a ramp for wheelchairs to get on and off trains , as with buses the bus is usually wheel chair accessible
you will need www.translink.co.uk
for any public transport journeys in Northern Ireland just type in Belfast Great Victoria street to " the aird the nook giants causeway " into the Journey planner this will bring up the train from Belfast to Coleraine (1hr 25 mins) and then you change in Coleraine and get the Ulsterbus 402 service to the giants causeway
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