When I ran your code inside vb6, it seemed to break a line in the middle of tblRequests, but still seemed to compile ok & give me a resulting string. Perhaps this may work (just added a couple of vbcrlf's)

Code:
Dim strSql As String
strSql = "Insert Into TblProgrammersHours (ProjectID,ProgrammerInitials, CurrProgHrs, PrevProgHrs, TotalProgHrs, Positions)"

strSql = strSql & " Select TblRequests.ProjectID, TblPersonnel.Initials, TblRequests.Prog1hrs, TblRequests.PrevProg1Hrs, TblRequests.TotalProg1hrs, 1 As Position " _
& "From TblRequests Inner Join tblPersonnel On TblPersonnel.[Name] = TblRequests.Personnel1 " & vbCrLf

strSql = strSql & "Union All " & vbCrLf _
& "  Select TblRequests.ProjectID, TblPersonnel.Initials, TblRequests.Prog2hrs, TblRequests.PrevProg2Hrs, TblRequests.TotalProg2hrs, 2 " _
& "From TblRequests Inner Join tblPersonnel On TblPersonnel.[Name] = TblRequests.Personnel2 " & vbCrLf

strSql = strSql & "Union All " & vbCrLf _
& "  Select TblRequests.ProjectID, TblPersonnel.Initials, TblRequests.Prog3hrs, TblRequests.PrevProg3Hrs, TblRequests.TotalProg3hrs, 3 " _
& "From TblRequests Inner Join tblPersonnel On TblPersonnel.[Name] = TblRequests.Personnel3 " & vbCrLf

strSql = strSql & "Union All " & vbCrLf _
& "  Select TblRequests.ProjectID, TblPersonnel.Initials, TblRequests.Prog4hrs, TblRequests.PrevProg4Hrs, TblRequests.TotalProg4hrs, 4 " _
& "From TblRequests Inner Join tblPersonnel On TblPersonnel.[Name] = TblRequests.Personnel4 " & vbCrLf

strSql = strSql & "Union All " & vbCrLf _
& "  Select TblRequests.ProjectID, TblPersonnel.Initials, TblRequests.Prog5hrs, TblRequests.PrevProg5Hrs, TblRequests.TotalProg5hrs, 5 " _
& "From TblRequests Inner Join tblPersonnel On TblPersonnel.[Name] = TblRequests.Personnel5 " & vbCrLf

strSql = strSql & "Union All " & vbCrLf _
& "  Select TblRequests.ProjectID, TblPersonnel.Initials, TblRequests.Prog6hrs, TblRequests.PrevProg6Hrs, TblRequests.TotalProg6hrs, 6 " _
& "From TblRequests Inner Join tblPersonnel On TblPersonnel.[Name] = TblRequests.Personnel6 " & vbCrLf

strSql = strSql & ") As NewProgHrsTbl"
Debug.Print strSql