Trim, LTrim and RTrim will not help you here - they will just remove the blank spaces from the beginning and the end of your string.
If "name" is always preceded by a hyphen ("-"), you can do the following:
Dim strFieldValue, intStartPosition, strName
strFieldValue = "Supplier - Name"
intStartPosition = InStr(strFieldValue, "-")
strName = Trim(Mid(strFieldValue, intStartPosition +1)) 'Trim here is used to cut the blank spaces before and after the name, if there are any.