Two easy and inexpensive options that should wrk for you would be stick-on vinyl tile and a floating laminate floor. They are pretty self-explanatory and you can get them at the big box stores and install them DIY.
While I agree with @ about the floating laminate floors (looks like wood, ceramic, stone), I would recommend you stay far away from the self stick tile when it comes to mobile home floors.Instead I would look at rolled laminate that doesn't need gluing & instead relies on tape at the edges.
A few quick items:
First most mobile homes do not have carpet installed like a traditional house. Instead of using tack-strip they roll it out throughout the structure, staple it in place & then build the interior walls over it. In order to remove it & not have issues later - angle your blade in at a 45 and follow the interior walls as you cut
No matter which product you install, you can't leave any dirt on the floors or other inconsistencies - the better the substrate, the better the finish
If you do go with a floating laminate floor, you will have to install baseboard as you need to leave a 1/4" gap all the way around for expansion & contraction (The directions on the back of the box are gold)
You probably should install a molding for the rolled version, but it is not necessary - if you are not going to make sure the tape follows the whole perimeter to help prevent it from curling up
For vinyl, If you have enough experience, you can easily cut it in place however if you lay out rosin paper in the room, trim it to shape & tape it together, you can easily use that as a template.One trick many use is to cut it about an inch or so short & mark it with a level placed against the wall. They then take it outside, place it over the vinyl & tape it down with blue painters tape. Then using the same level as a cutting guide they follow the lines & it fits perfectly when they roll it out
Another option is a free floating vinyl product like Traffic master Allure
I agree also with the others. But just understand that your floors will tend to feel much colder. All depends upon what the insulation is like under the floors. You may want to consider a radiant heat mat under the floating floors to keep your toes warm in the winter months.
If you decide to put down laminate, make sure you spend the few extra dollars and get the "quiet pad"! You won't regret it. I have laminate and LOVE IT! It has been down for 8 years and looks like new! I use a Swiffer wet pad to clean it! You do not have to take up the vinyl to put this down but will have to take up the carpet. We put down pre-finished quarter round. Where I had wood meeting laminate, I put down decorative thresholds. Make sure you take a sample of your boards with you when you shop for tile thresholds. I even have laminate in my bathroom. I admit I am careful with water because it could seep down to the sub flooring.
I would suggest before you put down any new flooring you should paint the floor with any mistake paint you have on hand. The saw dust board mobile home makers like to use is junk when it gets wet so if you paint it that will seal the wood so it go to junk if it gets wet. We own several mobile homes and we do that to all the rentals.