I think it depends on whether you are the person tendering, or the person inviting tenders.
Either way, where I work, the person tendering tends to be called the "bidder" (or tenderer of course.)
From the bidder's point of view, the organisation inviting tenders would be "the customer". From the other p.o.v, I'm not so sure. Perhaps the organisation would refer to itself as "the company" or just use its own name.
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