Compare data from Sheet2 to a summary table on Sheet1

I'm attaching an excell sheet with sample data and desired output in RawData

RawData contains every record where same name may be repeating multiple times.
KeyData has unique key to define the ID for each type

If Name in RawData sheet matches Name in Key Name column, then copy ID to RawData from Matching Key ID.

Closest I could get if there was a matching column,

=IFERROR(VLOOKUP(B2,Key!$B$2:$B$24,1,0),"No match")

However, when I try to expand it, my logic is lost somewhere

=VLOOKUP(IFERROR(VLOOKUP(B2,Key!$B$2:$B$24,1,0),"No match"),Key!$A$2:$A$24,1,0)

Thanks for help!

Thank you for a fast reply. Worked perfectly!