Putting a subwoofer in a media cabinet is possible and won't vibrate IF you read the fine print! I have a pair of subwoofers in my garage that shake the structure from sound pressure but the cabinets themselves don't vibrate. I use a PPSL enclosure or push-pull slot load to prevent vibration. Basically, one sub is mounted in phase with the cone pointing out as normal and the other sub is mounted magnet side out and wired out of phase. In operation, one cone goes out and the other goes "in" which cancels vibration from the drivers.
The other trick is to make the box heavy--I use 1.5" thick slot/top/sides and front to do this. The sub itself weighs 185 pounds with 8.7 cubic feet internal air space. Push-pull subs are commonly used in arena sound where they hang sub boxes and don't want vibration or inertia to make them move--good idea!
If you want to integrate speakers into the cabinet--don't! The best way to do that is to figure out what speakers you wish to use and make shelves and a space to place the speakers inside. This will allow you to aim them correctly to fit your needs now and adjust them later as needs and life make changes to your room. Give them about 4 to 6 inches of space to allow changing the angle or adding spacers to raise them if/when the need arises.
To stealth them in is easy--make a speaker grill to completely cover the space that the speaker occupies. Use grill cloth that matches the cabinet or it can be dyed to the correct color. Heck, you can get white grill cloth and have pictures printed on it and frame the thing to cover the speakers location.
Using the shelf idea with a grill covering it all not only allows for complete stealth while offering the ability to make critical angle and height adjustments--it allows for a very important factor for the beginning DIY'er... the speakers can be butt ugly and you'll never see them. If you have a decent enough size space to place them, it allows for modifications to the speaker by enlarging the cabinet etc. without screwing up the furniture.
Being successful in DIY speakers is like being successful in the bar scene...go ugly early. Your first build should be performance oriented so hiding them would allow you to concentrate on sound quality and setup and not worry about pissing off the wife. If you find DIY is a fun and rewarding hobby, then you can learn the woodworking aspect of your quest. Best to have that large grill covering up your learning experience--you can pull the grills off to show people your prowess after you gain...prowess. You can line the space you place the speaker with denim/polyfill/OC 703 to keep sound reflections down then slap the grill to cover.