Old-fashioned character sets such as ASCII and ISO-8859-1 do indeed lack the two characters necessary for Welsh: W with a circumflex accent and Y with a circumflex accent. While ISO-8859-14 does support them, its use is quite rare.

Yet again, Unicode comes to the rescue:

Ŵ Ŵ U+0174 LATIN CAPITAL LETTER W WITH CIRCUMFLEX
ŵ ŵ U+0175 LATIN SMALL LETTER W WITH CIRCUMFLEX
Ŷ Ŷ U+0176 LATIN CAPITAL LETTER Y WITH CIRCUMFLEX
ŷ ŷ U+0177 LATIN SMALL LETTER Y WITH CIRCUMFLEX

Unicode also offers an alternative concept of combining characters:
̂ U+0302 COMBINING CIRCUMFLEX ACCENT
Combining characters are generally diacritics which, when following any character, combine with it in a single character space. Thus:
Ŵ = 
ŵ = 
Ŷ = 
ŷ = 

As usual with Unicode, older browsers such as Netscape 4 don't support many characters. Get rid of your '90s browser and update to Mozilla or Internet Explorer.
You also need a font which supports these characters but modern operating systems generally support all but the rarest characters.