。
She has been asked to become a member of the committee.
Literal
She [topic-は] that committee's one-member becoming [that-way-ように と] request [object-を] has-been-receiving.
~ように と~を受ける wraps an indirect request around a request-receiving verb. ように turns 一員になる ('become a member') into the content of the request, と marks it as quoted instructional speech, and 依頼を受ける ('receive a request') frames her as the recipient. 一員 ('one member') is a slightly formal way to say 'a member' — heightening the bureaucratic flavor that fits committee work. The ~てきた (or ~てきている) on the verb chain captures a process unfolding up to the present: the requests have been arriving over time.