Sum of a column based on name and date entered on another tab

There are lots of options but assuming you have 365 or at least Excel 2016 or newer you should be able to use SUMIFS:
=SUMIFS('Billing'!$F$1:$F$100, 'Billing'!$B$1:$B$100, ">="&B$3, 'Billing'!$B$1:$B$100, "<="&EOMONTH(B$3,0), 'Billing'!$G$1:$G$100, $A$4)
but better yet would be if you 'Format as a Table' the data on the Billing tab (and name the table 'Billing') so it might look more like:
=SUMIFS(Billing[Value], Billing[Date], ">="&B$3, Billing[Date], "<="&EOMONTH(B$3,0), Billing[Name], $A$4)

So 'Format as Table' can be found on the home tab, just click in the data area or highlight all the data and click that button and it suggest the area and make sure to include headers and select the formatting.  Then the 'TABLE' tab will come up and on the left side you can enter a name for the table.  Then to refer to a all the data in a particular column of the data Table you use the table Name followed by bracket of the column name like BILLING[name] to return the column of the table listed under 'name'.  Makes reading the formulas much easier and will include all the data in the table without referencing the entire column of the data sheet.