9th Gen CPU's have been supported as of BIOS revision 7B22v24, which released (to the public) in August 2018.
Unless you find a motherboard that's sat on a shelf for ~18months since being produced, it
should have a BIOS that is capable of supporting the 9100F out of the box.
I don't believe there was any underlying code update required to support the "F" variants of CPUs when they came out.
NOW, whether the i3-9100F + B360 is the best option at it's pricepoint (~$173 total), is debatable* - but no reason it shouldn't be functional.
*As an example, if you haven't already bought the board or CPU, you can have a 6core/12thread Ryzen5 and an entry level B450 board for very, very similar money:
PCPartPicker Part List...