Yes, the vast majority of chewable tablets can be swallowed intact with no problems. Chewable tablets are designed and manufactured to be broken up in the mouth, to aid people with swallowing problems or to increase palatability. When chewable tablets are swallowed whole, the acid in our stomach will dissolve them.
It is extremely rare for a chewable tablet that can not be swallowed whole, but there are examples. One such example is the prescription drug Fosrenol (lanthanum carbonate). After the drug was released, there were reports of serious GI obstruction and fecal impaction when chewable tablets were swallowed whole.
The above is a rare example however. In most cases, you can feel safe swallowing chewable tablets whole if you desire. If you have a question about a specific product, consult your pharmacist or check the package labeling.