Zoroastrianism does not forbid conversion, it was banned during the Muslim era, but only Indian Parsis kept this ban upon entering India (ethnic elitism)

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Does the religion prohibit accepting those who wish to join it ?

For the first few thousand years of its history, the religion welcomed anyone who wished to join it but prohibited forced conversions.  In the Gathas, Zarathushtra asks for Wisdom's help, so that he "... might convert all the living." Y31.3.  Speaking of efforts to spread Wisdom's teachings, he says,

"...the family, the community together with the clan, entreated for the grace of Him, the Wise Lord, (saying:) 'Let us be Thy messengers,...' " Y32.1;

"Lord, let wisdom come in the company of truth across the earth!..." Y50.5.

An ancient prayer, composed more than 1,000 years after Zarathushtra says,

"Homage to the all knowing tolerator [Tolerant One], who sent through Zarathushtra ...  teachings ... for the people of the world so that they may have friendship, inculcate ... inner wisdom and knowledge gained from hearing.  For the information and guidance of all men who are, who were, and who will be hereafter ..." from the Doa Nam Setayashne Sethna translation.

Another ancient prayer composed in the same era says,

"... May the religion of Zarathushtra be a source of glory unto all mortals..." from the Doa Tandarosti, Sethna translation.​

But forced conversions are forbidden.  An ancient text cautions, that if a man converts to the religion, he should not force his wife to convert.  He should not stop supporting her.  Her freedom to choose was respected.

After the Arab invasion of Iran in about 650 CE, Zoroastrians were persecuted almost to extinction, and accepting anyone who wished to convert from the dominant religion was punishable by the death and destruction of all concerned and sometimes their families, and communities as well.  So to survive, Zoroastrians adopted a policy of non-acceptance.  Those Zoroastrians who fled to India, took this tradition with them, and it continued to be practiced in India long after the reason for it  no longer existed.   But many Zoroastrian communities in diaspora are reverting to the original teaching of accepting those who wish to join the religion.