How do I check if a char is a vowel?

Actually there are much more efficient ways to check it but since you've asked what is the problem with yours, I can tell that the problem is you have to change those OR operators with AND operators. With your if statement, it will always be true.

Your condition is flawed. Think about the simpler version

z != 'a' || z != 'e'

If z is 'a' then the second half will be true since z is not 'e' (i.e. the whole condition is true), and if z is 'e' then the first half will be true since z is not 'a' (again, whole condition true). Of course, if z is neither 'a' nor 'e' then both parts will be true. In other words, your condition will never be false!

You likely want &&s there instead:

z != 'a' && z != 'e' && ...

Or perhaps:

"aeiou".indexOf(z) < 0

Clean method to check for vowels:

public static boolean isVowel(char c) {
  return "AEIOUaeiou".indexOf(c) != -1;
}

How about an approach using regular expressions? If you use the proper pattern you can get the results from the Matcher object using groups. In the code sample below the call to m.group(1) should return you the string you're looking for as long as there's a pattern match.

String wordT = null;
Pattern patternOne = Pattern.compile("^([\\w]{2}[AEIOUaeiou]*[^AEIOUaeiou]{1}).*");
Matcher m = patternOne.matcher("Jaemeas");
if (m.matches()) {
    wordT = m.group(1);
}

Just a little different approach that accomplishes the same goal.