Does tai chi involve any strikes in its system?

It just seems to be nothing but pushing and pulling is there any punches or kicks at all in Tai Chi???

It is all in there. Strikes, punches, palm strikes, grabs locks, pressure points, leg techniques, ....etc. it is all there but not obvious.

Don't mistake the push hands exercise as the way a Tai Chi Chuan practitioner would utilize his/her training in a real self defense situation.

Push hands is an exercise about learning to deal with incoming force. A push, accellerated to a faster speed, can be turned into a strike. Some of the stepping in the form is to allow for a low line kick to be inserted. Some of the movements are actually set-ups for grappling moves or throws.

The best thought I ever heard about the internal arts is that the forms teach movements. Ideas of what form the body can take. The principles taught are a reminder to be balanced, centered, etc. so that one can use the movements effectively.

Never judge a book by its cover.

Practically every martial art technically has all aspects. The problem is that people judge arts by what they specialize in, what they learn first, and by what they see in competition. Tai Chi focusses on pushing, pulling and redirection, but it contains strikes, kicks, locks, throws, and pressure points, along with Chi Gong and energy work.

It's all in there, punches, kicks, throws, grappling. The combat forms of Tai Chi are essentially a complete martial art so elements of everything are of course included.