No, the Vedas don't declare Shudras to be eligible to recite the Vedas. If they would be eligible for Vedic study, then the Taittirya Samhita of the Krishna Yajurveda wouldn't forbid Shudras to perform Yajnas:
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. (Taittirya Samhita 7.11Taittirya Samhita 7.11)
So if we interpret Shukla Yajurveda 26.2 to be allowing anybody to study the Vedas, it is basically contradicting the above verse which is also from the Vedas.
Therefore, the more appropriate interpretation of Shukla Yajurveda 26.2 would be that Ishwara is asking people who are eligible to study the Vedas i.e. Brahmanas, Kshatriyas and Vaishyas, to spread the meaning of the Vedas to others. That doesn't necessarily mean Shudras, Mlecchas and Avarnas can hear or read the Vedas. It is basically promoting the study of Vedas to people (who are eligible) not only for their benefit but of mankind.
For example, if someone performs a Yajna for rain which comes as a success, it is not only that person who will benefit from the rain but all sections of the society.
It's like "if you water the roots, the leaves and branches grow automatically". So by teaching eligible people Vedas, the benefits can be reaped even by people who are not eligible.
Vedic mantras are supposed to be highly classified and secretive knowledge. So one must have the necessary qualifications to study them. It should not be revealed to all.