The short answer would have to be, yes, you definitely can. There are certain cases where it’s not a desirable solution though. The factors that may affect the quality of the newly installed bamboo over a vinyl tile on plywood floor may vary from one house to the next, so you need to take your time and figure out whether doing it in your place is conceivable.
To start with, the main task of a plywood floor is to act as an underlayment for the vinyl tile in the first place. That means that if it’s been properly installed, it’s an ideal smooth, clean, protective layer that prevents moisture from affecting the floor above it, as well as give an uniform constructional integrity across the room. In other words, if your plywood floor is in good condition, you can use it as it is and rest assured your bamboo floor is provided with a high quality insulation.
The problem arises from assessing the plywood floor properly, since you have the vinyl tile on top of it. It is best if you have some background information about the plywood and vinyl tile installation job, but that is not always possible, especially if someone else worked on that. In those cases you have only two options. Either rely on the good condition of the vinyl tile to support the bamboo floor well, or remove the vinyl and plywood altogether.
Since the latter is not the aim here, you need to observe the vinyl tile condition and make decisions based on that. It may very often give indirect clues for the plywood sub-floor as well. A well preserved, even, dry and smooth vinyl tile material speaks of high quality protection underneath, as well as good maintenance. It can act as an ideal sub floor to the bamboo planks and combined with the plywood, there is probably no better way to protect a floor without doing massive renovation work.
If the vinyl has some damaged spots, it’s possible to fix them to some extent. Irregular surfaces can be smoothened out with filler substances often used on floor installation sites and which you can find in most construction shops. As long as the vinyl is well adhered to the plywood underneath and there are no large undulations or bumps anywhere along its surface, it can be prepared to work as a bamboo under floor well enough.
One thing that’s important from a safety point of view is the possible use of adhesives for vinyl tiles that contain a high level of asbestos. A dangerous substance that can easily contaminate your house if you go on to remove damaged parts or the whole vinyl tile floor. For that reason it is a job best left for professionals who you should rely upon if you’re not certain of the type of adhesive used for the vinyl floor cover. You can also carry out a test that can reveal the level of asbestos contained in the adhesive and make any further decisions based on that.