I have a 14 year old Homeline 200 amp/40 circuit service panel that I would like to install at least 2 to 4 tandem breakers in. When I went to press one of the tandems that I had purchased at Menards into place, it would not go in all the way onto the bus. Upon further inspection I realized that the slot was shallower in the tandem than was the case with the original breaker that I had removed. It would be really easy to modify the breaker so it fits, but I've been told that would not be wise to do. Given that limited amount of information, do they make a tandem breaker that is designed to fit into my service panel, one that would also allow the cover to fully close without modifying the cover?

No.

A 40 space Homeline panel will not accept twin breakers. Modifying either breakers or the panel buss are NOT good ideas.

Your best bet will be to install a small sub-panel to gain some breaker space. You can move a few of the circuits from the main panel over to the sub to create some space for the sub-feed.