4094 IC Strobe and Clock

If I connect STR and CLK together on the 4094 and configure things as a regular shift register, will it work?
The point of my circuit is just to be a serial to parallel shift register with only CLK, DAT, and OE inputs, disregarding the others.

Why do you want to clock the STR pin? If you do that it will likely operate as if the strobe is tied low.

Just tie the STR line high and it should do what you want.