Agree with some of the prior posts.
The vinyl could be old which can cause the adhesive to stick to the release liner more.
A squeegee is nice but it isn't needed. You should be able to get the vinyl to adhere to the transfer tape just by rubbing it with your fingers. It doesn't take much pressure to create the bond between the transfer tape and the vinyl.
There are several different types of transfer tape. Low tack, medium tack, high tack. If the transfer tape isn't pulling the vinyl off the release liner it is possible you have a low tack transfer tape. For what you have perhaps a medium tack or high tack transfer tape is needed.