I am not aware of a direct call in AdWords API, although the new Ads API (currently in beta) may have some.
I do know a workaround. Identify API calls that fail when the user has no sufficient access rights.
Then call that exact API call with header flag "validateOnly" set to true. When that flag is set, the API will report errors, but if no errors occur, the API will NOT execute the call.
The idea is that you check whether the error occurs. However, be aware of different errors.
For instance: You could try creating a campaign with validateOnly. When that succeeds with validateOnly set, you know that the user has more than read-only access. But, when it fails, you should be aware of different errors as well, for instance you didn't supply a campaign name, or its name was too long, or a campaign with the same name exists, etc. In other words, on failure you've got to examine the error.
A special warning about disabled accounts (not campaigns or groups, but accounts): when an account is disabled, all API calls fail. Reasons for Google to disable an account include (for instance) lacking payments, repeatedly violating T&C, etc. But, when you think about it, that's a special type of access, so it does make sense.