If you are looking for the births of Ella Fulton and any siblings then GRONI is the only repository for birth, death and marriage records in what is now Northern Ireland, from 1.1.1922 onwards. (Ancestry and other websites don’t have them).
Births in Northern Ireland within the past 100 years, marriages within the past 75 and deaths within the past 50 aren’t searchable on-line. They are open to the public, just not on-line. If you know all the key details eg exact date and place of birth, parents names etc, you can order a copy of a certificate from GRONI for £15 (sterling). https://geni.nidirect.gov.uk
But if you don’t have all the information, you need to get in touch with them to get a search done. They will do a 5 year search for £7 and if they find a matching certificate you can purchase it for £8, coming to a total of £15. It will be posted to you a few days later. You pay over the phone using a credit card. You type your number in on the phone keypad. They don’t know what it is. They won’t deal with these searches and orders by e-mail or post because they need you to type the credit card number in yourself.
Phone: 0300 200 7890 (+44 300 200 7890 from outside the UK)
Alternatively if you live near Belfast, you can go in person to either GRONI or PRONI (both in Belfast). They are the only places where you can search recent records, right up to those registered yesterday. You can view each cert there for £2.50, and if you want a certified copy it’s an extra £8. (Posted out in 2 to 3 weeks time).
I see 3 Arthur Fulton marriages in NI in the 1920s:
3.10.1924 in Belfast spuse surname Magill
14.4.1925 Strabane spouse Jamieson
26.12.1929 Belfast spouse McWhirter.
I’d guess the Strabane marriage is the most likely. You can view that on-line on the GRONI site for £2.50.
John Fulton married Margaret Hannigan in 1882 in Cappagh:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1882/10967/8008730.pdf
I see the following children to that marriage:
Catharine 16.3.1883
Arthur 13.1.1885
Sarah 14.2.1887
Anne 22.9.1889
John 9.11.1891
James 13.2.1894
The mother Margaret looks to have died in 1897. Daughter Catharine was the informant:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1897/05874/4665539.pdf
John Fulton looks to have remarried a few months later:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1897/10462/5809830.pdf
Family in 1901:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Moyle/Stralette…