Yip - use the VLOOKUP function.
In spreadsheet A, column C (next to your reg numbers and mileages) insert function VLOOKUP.
For Lookup Value it will be the first cell in column A containing a reg number.
For Table Array it will be the the range of your data in spreadsheet B.
For Column index number it will be the number of the column in spreadsheet B that contains the driver's names (A=1, B=2, etc)
For Range Lookup type in False to get an exact match.
Now copy the function down column C of spreadsheet A.
That should do the trick for you.
BW