Where does the system load go on a bq51051b charge controller?
The bq51051b is a wireless battery charging solution, so its only really meant to charge batteries, if you wanted to though, I see no issue with just hooking up your load in parallel with the battery while its connected



This image is from section 9.1 of the datasheet.

Please note that this IC doesn't have any over/under-charge protection.  You'll need either another IC, circuit, or a battery that has the technology included.  This chip does, however, have over-current protection on the charge path.  You'll need to properly calculate the the resistor on the ILIM pin in order for this to work.  The formulas are in the datasheet.