Chat assistants increasingly integrate web search functionality, enabling them to retrieve and cite external sources. While this promises more reliable answers, it also raises the risk of amplifying misinformation from low-credibility sources. In this paper, we introduce a novel methodology for evaluating assistants' web search behavior, focusing on source credibility and the groundedness of responses with respect to cited sources. Using 100 claims across five misinformation-prone topics, we assess GPT-4o, GPT-5, Perplexity, and Qwen Chat. Our findings reveal differences between the assistants, with Perplexity achieving the highest source credibility, whereas GPT-4o exhibits elevated citation of non-credibility sources on sensitive topics. This work provides the first systematic comparison of commonly used chat assistants for fact-checking behavior, offering a foundation for evaluating AI systems in high-stakes information environments.