Laura,
William and Sarah’s marriage and the births of their children are all before the start of statutory records in Ireland (1864 for births and 1845 for non RC marriages). That means you won’t find birth certificates or a marriage certificate for this couple. You might still find those events in church records but those records may not be on-line.
You haven’t said what denomination the family was but judging by the location, and the Scottish surnames I’d guess they were Presbyterian. (Three quarters of the population of Ballymena was Presbyterian, descended from Scots who settled there in the 1600s). If so, there are about 4 Presbyterian churches in Ballymena with records for the 1830s and 1840s. 1st Ballymena Presbyterian has records from 1812, 2nd Ballymena from 1813 & West Church from 1829. You could also try the Church of Ireland which has records from 1789 (with some gaps).
There are copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
You’ll probably struggle to find much information on this couple (apart from the church records). If William and his family’s fare was paid by a cousin, you can probably infer he wasn’t very wealthy and was likely a labourer, either working on farms near Ballymena or in a mill or similar in the town itself. There’s a link here to Slater’s Directory for 1846 which lists the key tradesmen in the town. William isn’t listed as far as I can see, so that too point sot him being in some labouring role.
https://www.failteromhat.com/slateru.htm
