You can install carpet right over vinyl tile or any other type of flooring. Carpet will install easily with really no extra work than there would be installing over a wood floor surface. The only problem may be water seeping under the carpet and staining it. It also may be a little more difficult than installing carpet over padding.
Carpet over Vinyl
Preparing to Install
The first thing to do is to clear the floor and prepare to install. You should have a carpet piece that fits the size of the room. The next step is to install carpet runners around the edges of the walls. These can be glued right to the vinyl or tacked into the floor if it will take a nail. The other option is an all-adhesive install.
Gluing Carpet Down
To install over vinyl you can also glue the carpet down. This is not recommended if you think you may want to change out the carpet at another time. To glue down carpet, choose an adhesive that will hold fiber or cloth to vinyl. Then, using a caulking gun, create a bead of this adhesive around the edges of the room and then spread long wavy lines of adhesive across the floor. Start in center and work toward the exit when doing this so you don't have to walk back across the glue. Then install the carpet to the vinyl floor piece by piece by bringing it back through the doorway and laying it down in front of you so you have something to walk on.
For better results, you can use a carpet pad under the carpet. This pad should be laid down right over the vinyl but not over the carpet runners. The carpet just goes right over the top of the pad. You should not glue the pad and carpet together and then glue it to the vinyl as this would cause too much work if it ever needed to be removed. You do not need to remove foot molding to install carpet over vinyl