首先fall on中on一般为介词，例如:Her birthday will fall on a Friday this year.在你这句话里应该直接用falls。
其次fall，一般用于说庆典或是特殊节日于某日发生，这里season我认为用starts或是comes更好一些。
然后就是定语太长了，建议放在后面。修改后是:The season starts when our drama club aborbs new members.

我好奇一下，社团招新用absorb是不是从中文“吸收新成员”硬译过来的？母语人日常说这句话没那么复杂，就简简单单It's recruitment/recruiting time for our drama club就行了

He tapped me on the shoulder, 'It's recruitment season again.'
He caressed the chloroform soaked handkerchief, like he did that girl last summer. 'It's time to lure another in. How dramatic.'

The season for our drama club to recruit new members comes.

接纳，吸纳，induct
示例
The fraternity will hold a ceremony to induct the new members. 兄弟会将举行仪式接纳新成员。

我高考英语刚及格，刚刚念了一遍感觉不对劲，像是没有谓语