I have a spreadsheet which I use to track the jobs we do every month.
My manager has asked me to modify the sheet to track billable hours.
The amount we can bill will vary depending on who is doing the work.

There is already a box on the spreadsheet next to each job in which we
record the person to whom that job is assigned. At the top of the
spreadsheet I'e created a table, which I want to use to track who
is working what billable hours. I've now added another column so we
can enter the number of hours worked on each job.

Can I use IF statements to track who is working what hours?

So for instance could I enter a formula into one of my table's cells
that says something like this:

'Look at cells B5 to B12. Every time you see the initials RM in one
of these cells, look at the corresponding cell in row C. Add up all
the values in these cells in column C.'

Looks like a job for SUMIF. An example is in the CSV file below.

A different approach is a pivot table report. It's described in Excel's
built-in Help.

----------------------------- cut here -----------------------------
,,,RM,"=SUMIF(B:C,D1,C:C)"
,,,BV,"=SUMIF(B:C,D2,C:C)"
,,,DA,"=SUMIF(B:C,D3,C:C)"

1,RM,1,,
2,BV,3,,
3,BV,1,,
4,RM,5,,
5,DA,2,,
6,BV,7,,
7,DA,2,,
8,DA,9,,