How important is it that I use a 4069UB
The important bit is the "UB" after "4069". It stands for unbuffered and, an unbuffered device is needed. A 7400 is "buffered" i.e. it won't work as intended.
Is there another way of making an 8MHz oscillator with the crystal and some 74xx logic gates?
Providing it inverts and is unbuffered then it should work. Use CMOS parts though because TTL devices will have input impedances that are too low to get reliable oscillation through a crystal.
You might also add this resistor here: -
