Can I use a ULN2003A IC as many transistor switches?
Question
Can I use ULN2003 to switch 5V relays which in turn switch 220VAC loads?

Answer
No problem at all.
Your ULN2003 choice is good, because it can handle max 150mA, and the most popular, typical 5V relay switch, Songle relay switch, takes only max 70mA.
I usually prefer ULN2803 which has 8 open collector, power NPN BJTs, ideal to sink 5V Songle switches.

Discussion, Conclusion, and Recommendation

You may like to also consider I2C 16 channel MCP23017 GPIO expander. Using 2 I2C wires/signals you can control 8 MCP23017 for 8 x 16 relays.  Another good thing aboue MCP23017 is that you can set input or output, interrupt control, and the output pins are sort of latched, ie, once setup, no need to take care afterwards.

For prototyping or feasibility studies, you can buy cheap ULN2803 and MCP23017 modules which save soldering effort/time and space.


References
(1) MCP23017 I2C 16 Bit Bidirectional I/O Expansion Module - US$0.75 ~ 1.28
(2) What is an open collector?

Appendices
Appendix A - ULN2003/2803 Wiring Diagrams