Is there a way to make the adapter only work in a ready only mode, perhaps in the connection string, for the Access DQE?

AFAIK no. Please elaborate more why you want such thing.

I have the difficult task of creating a web-app based on Access and I thought since I am executing read-only commands telling the Adapter that it will never write might lighten the load?

SQL Server would be better , but I am limited to Access only.

What do you mean exactly with 'lighten the load' ? The adapter just creates a connection and fetches, it doesn't open transactions for fetching if you don't ask it to so if you don't save anything, nothing is different from what a readonly adapter would look like. The rest is just code and that doesn't affect performance.