Can I store documents like proposals on a contact and send them to the contact by email?

Yes. Inside a contact there is a Files feature where you can add any kind of file to that contact, such as documents, proposals, estimates, contracts and logos. From there you can also open the media library, which is the same as the Library page where all your files are.

Every file can get a shareable link. Put that link in an email (for example: here is the link to your proposal) and the person can open the PDF. Emails can be automated, so the link can also go out automatically from a workflow.