I clicked the inquiry email link and nothing happened
This is expected — mail scanners open the link automatically before you ever click it, so the link itself does nothing. Open the confirmation page from the email and press the button on it; that button is what actually confirms your inquiry.
Can a foreigner buy a condo in Phuket in their own name?
Yes — a foreigner can own a condo in Thailand freehold, in their own name, as long as the building still has room in its 49% foreign-ownership quota and the purchase funds arrive from abroad in foreign currency (the FET form).