Most coffee shops currently are closed due to temporary social distancing orders. However, there are some places offering online ordering. I have not personally tested all of these, but according to information provided on the websites, their online and subscription services are still operational (taking into account possible delays in the postal service).
Workshop Coffee offers a straightforward subscription service and ship coffee from their roastery in Bethnal Green. You can choose delivery for every week, fortnight, or month all at £12 per delivery of 250 g. You can pause or cancel the subscription at any time.
The shop at Borough Market is still open for business (as of the week of 20 April), but the website offers an excellent subscription service with a choice of whole or ground beans over a period of 3, 6, or 12 months.
Kapihan is inspired by the coffee culture of the Philippines. The location in Battersea is temporarily closed until restrictions are lifted. For only £12.50 per 250 g the coffee is available to order online.
Hundred House Coffee currently is offering free shipping in the UK when two bags (227 g) or more are ordered. The site offers a variety of choices and subscription options with the possibility to pause or cancel with no commitment.