You need to be more clear what you need here.
It sounds like you have created an Access application, with Forms, Reports, etc. While you could link from an Access "front end" (which would need to be installed onto each client system) to a SQL Server, etc. "back end" located
in the cloud I doubt you can just shove your existing Access application onto some cloud sevice.
Whether using Jet or ACE as your current datastore, neither is practical to put on a cloud service. These require remote filesystem access between the client piece and the datastore, and performance would be pretty bad. You'd need to move to a client-server DBMS backend.
Of course if you have a monolithic "Access application" they want to download and run locally, well then you just want a file sharing service. Which (despite the marketing hype) isn't really "cloud" anything.
If they want this to be more broadly available you might have to convert it to an Access Web application, which just becomes a sort of web app, and again: no "cloud" involved.
Could be Office 365 might offer you something, see
and/or
However there is no "cloud" here either.
Just sticking something on a remote server is not cloud computing.