Skimming over it, the implementation looks inefficient:

    int matchstar(int c, char *regexp, char *text)
    {
        do {    /* a * matches zero or more instances */
            if (matchhere(regexp, text))
                return 1;
        } while (*text != '\0' && (*text++ == c || c == '.'));
        return 0;
    }

That can be used to match short strings, but not to grep through a filesystem. A good regex matcher has runtime O(N*M) where N is the length of the regex (typically very short) and M is the length of the scanned text.