Re "undergraduate house". My own college's Career Development Office offered to keep letters of recommendation on file in case we ever needed one after we graduate; but I don't know a single student who actually utilized that service while applying to graduate school. In the US, professors are used to submitting letters to every single graduate program themselves. If you are in a different country and concerned about being too demanding on your references, I would suggest that you have an open conversation with them.
For example, when I was applying to REUs (with all applications done on paper), I gave my professors addressed envelopes, so that they only needed to print and sign a bunch of letters, stuff them in the envelopes and hand them off to the department secretary for mailing. (If your professors don't get free mail through the department, you should probably put stamps on the envelopes too.)
If your professors don't mind you seeing the letters, I guess you could stuff the envelopes yourself - but that should be their suggestion, not yours.
I personally would not want to do graduate applications on paper though. The big advantage of online letters is that you can track them - you know exactly what's been submitted and what hasn't. With paper letters, there's the insecurity of whether it's actually been mailed, if it might have gotten lost in the mail, and if the department filed your letters correctly.