Can Shudras do Upanayanam or perform Yajnas as per Vedas?
No, the Taittiriya Samhita of the Krishna Yajur Veda forbids Shudras from engaging in sacrifices.
From his feet he meted out the Ekavinça Stoma. After it the Anustubh metre  was created, the Vairaja Saman, of men the Çudra [Shudra], of cattle the horse. Therefore the two, the horse and the Çudra, are dependent on others. Therefore the Çudra is not fit for the sacrifice, for he was not created after any gods. Therefore they depend on their feet, for they were created from the feet.
Also, note the phrase "for they were created from the feet", is found in the Purusha Sukta hymn as well.