We study whether large language models acting as autonomous bidders can tacitly collude by coordinating when to accept platform posted payouts in repeated Dutch auctions, without any communication. We present a minimal repeated auction model that yields a simple incentive compatibility condition and a closed form threshold for sustainable collusion for subgame-perfect Nash equilibria. In controlled simulations with multiple language models, we observe systematic supra-competitive prices in small auction settings and a return to competitive behavior as the number of bidders in the market increases, consistent with the theoretical model. We also find LLMs use various mechanisms to facilitate tacit coordination, such as focal point acceptance timing versus patient strategies that track the theoretical incentives. The results provide, to our knowledge, the first evidence of bidder side tacit collusion by LLMs and show that market structure levers can be more effective than capability limits for mitigation.