No, you can't do this without infringing the Mets' trademark. The MLB would very likely sue you, or file a "UDRP" action against you to get the domain name from you as a bad faith use of their TM. They'd claim that your site causes consumer confusion, because consumers might go to your site and buy your products thinking that they're official MLB products.
And why would you sell products on a site called "report" that would seem to be informational, rather than a retail outlet? That's misleading as well. But if you sold unoffical/counterfeit Mets products or re-sold actual, licensed, official, Mets products, that would also be trademark infringement and unfair competition.
Use of the Mets logo is irrelevent, since the team's TM is a typed "word" mark, not a logo.
But what you might be able to is start a "gripe" website, like newyorkmetssuck, which criticized or commented on the team, but if you sold retail products on it, you'd destroy what would otherwise be a good 1st amendment defense to a claim of TM infringement.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Q: Can I start a website with the name of a sports team?
R: Maybe, depending on what the domain name is and what you publish at the website.
Q: For example, can I buy and use the domain name "NewYorkMetsReport.com"
R: Maybe. It depends on what you publish at newyorkmetsreport.com.
If by "report" you mean information about the team, its players, its game results, etc. and you then publish such content [of your own creation], then using the trademark "new york mets" in that domain name is quite likely lawful.
Q: ... can I then sell anything on the site?
R: Maybe, it depends on what you sell.
Whether it's lawful or not to use "new york mets" in a domain name depends on whether consumers, after viewing the newyorkmetsreport.com homepage, would likely be confused into falsely believing that the New York Mets organization or Major League Baseball sponsored or endorsed the site. If yes, you can't lawfully use the domain name; if no, you can.
The two most respected appellate court justices who write about intellectual property, Justices Posner and Kozinski, have both penned a majority decision in their Circuits on this very issue and both concluded that the retailer's use of a famous trademark in its domain name was lawful [bargainbeanies.com (see http://j.mp/cOPe0z ) and buyorleaselexus.com and buy-a-lexus.com (see http://j.mp/aAq3nw )]. The second case is factually more analogous to your newyorkmetsreport.com example.
All of which is to say that MAYBE your use of newyorkmetsreport.com would be lawful. You need to speak with a trademark attorney about what you can publish at that site and what you can offer to sell via the site.
As a practical matter, even if your own attorney decides your use of "new york mets" is lawful, the New York Mets organization or Major League Baseball can still sue you or initiate a domain name dispute proceeding. Which would cost you a lot of money to defend. So that's a real risk -- which you may not to incur. Isn't business fun?
You need to avoid confusion in the market place. There are many factors that go into determining if a mark is confusingly similar to another, including the goods or services of the respective users, the market places in which the marks and good are used and the like. The Courts have set forth a multi-prong test to make such a determination given the facts of a particular situation. Generally, if the marks include the same formatives and the goods travel in the same markets, care must be taken to avoid infringing the earlier user's rights in their intellectual property.
Since your position may be quite expensive to defend, as Major League Baseball (MLB) will likely defend their mark "New York Mets," you should get the permission of MLB should you decide to proceed. In the alternative, t may be more cost effective to choose another domain name. It would be wise and cost effective to consult an intellectual property attorney for advice on how best to proceed.
Good luck in starting and growing your business.
Disclaimer: This answer does not establish an attorney-client relationship and does not constitute legal advice. It is for general information purposes only.