回复：据说在劳工部主页可以查看公司是否赞助H1B,

not 公司是否赞助H1B, but a list of companies that had ever filed an LCA in the defined period of time. Law clearly says no employer is obligated to sponsor visa.
Some companies may have filed LCA in the past, but no longer willing to sponsor H1 anymore. Or it sponsored H1 for person A, but it can refuse to sponsor person B. it's totally up to the company.