Shudras are not allowed... Whether, even the Kshatriyas or Vaishyas are also allowed, is something that is unclear, if we go by the minor Upanishads called the Sannyasa Upanishads.
The Narada Parivrajaka Upanishad (NPU) and the Brihat Sannyasa Upanishad (BSU), both discuss, the question- Who are all fit to renounce?
Thereupon Narada asked the Grandfather: "Lord,  how does a man
renounce? And who is entitled to renounce?"
"I shall first explain who is entitled to renounce and then describe
the procedure of renunciation. Listen attentively. "The blind,
eunuchs, outcastes, cripples, the effeminate, the deaf, the
childish, the dumb, heretics, discus-bearers, phallusbearers,
Vaikhanasas, Haradvijas,10 mercenary teachers, bald men, and people
without a sacred fire: these, even though they are detached, are
unfit for renunciation. If they have already renounced, they are not
entitled to receive instruction in the Great Sayings.11 An elder
renouncer12 is entitled to become a Paramahamsa.
So, Sudras are clearly not allowed, as they don't have the Vedic fire and likewise the Vratyas are also disallowed..
Similarly, from BSU Chapter2:
Sanctified by the forty sacramentary rites, he becomes indifferent toward everything; attains purity of mind; and burns away desire,
jealousy, envy, and egotism.  Only such a man, endowed with the
four aids, is fit to renounce
One should never admit to renunciation apostate renouncers, mortal
sinners, Vratyas, and the infamous.3 One should never admit to
renunciation those who have neglected vows, sacrifices, austerity,
liberality, fire offerings, and Vedic recitation, and those who
have fallen from truth and purity.
Now, Sudras do not have all the 40 Samskaras, they are not allowed to recite Vedas either. So, they are clearly not allowed..
Vratyas (i.e who are born in Dvija families but did not receive sacraments within prescribed time) are also similarly not allowed.