Can I use both I2C and SPI at same time with PIC16F1503
Yes, you can use this part to drive both a IIC bus and a SPI bus in the same circuit.  However, you can't (reasonably) use the single MSSP module of this device to drive both busses.

My recommended solution is to use the MSSP to do IIC and do the SPI by firmware bit-banging of separate I/O pins.  SPI is a very easy protocol to implement when you are the master, which it appears you are.  Since the master owns the clock, there is no problem if your SPI code gets interrupted and arbitrary delays inserted thereby.  Take a look at how SPI works.  It really is very simple and easy to implement directly in firmware.

You could of course do the IIC in firmware and let the MSSP manage the SPI bus.  I have written both, and SPI is easier to do in firmware than IIC, so I'd let the MSSP take care of IIC.  IIC isn't hard, but it is a little more tricky to do robustly.