Replacing shift register with PIC

[skimask]

I was wondering if it is possible to replace a 4021B (parallel to serial) shift register chip with a PIC using shiftin and shiftout functions?

I don't see why not. I think the biggest problem would be how fast the incoming data and outgoing clock signal would have to be.
But yes, 8 data lines and a latch signal for an input, easy enough... Put the data on the lines, watch for the latch to go low (or high, whatever), grab that data and store it, wait for the latch line to go back high (or low) and start waiting again.
Transfer that data to an output line using either shiftout or the builtin MSSP, easy enough also. If you use shiftout, the PIC would be tied up shifting that data out, if you used the MSSP, you could theoretically clock it out at about 2.5Mhz, depending on the PIC and clock speed used.