There are 2 telecommunication service providers in Brunei: DST and progesif. DST is the market leader and has the best coverage.
You can purchase both SIM cards at any operator store. You have to pay an annual SIM card license fee of B$25 to the government. You can avoid paying the fee by getting a tourist SIM card. I got one from DST (Easi tourist SIM card). It cost me B$15, comes with B$5 worth of credit and is free of the license fee.
Data cost on Easi tourist SIM card: B$ 0.20 per MB
Where to buy SIM cards
I got mine at Brunei international airport. There's a SIM card store on the ground floor of the airport mall. I think the store is open every day from morning to 5 p.m
Outside the airport, you can use branch locators to find stores for both DST and progesif.