The general rule is that you may not publish another company's logo on your website without written permission [that is, a license] from that company. There are exceptions that only your own trademark attorney can evaluate bases on your particular circumstances.
The above response is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
You may not necessarily need a formal license agreement. A simple written permission (via email) might be enough, depending on the nature of your use and the policies of the company. I would explore your options with an attorney. Legal issues aside, it is also probably a good business decision to speak with the company before using its logo. One way to avoid the issue in the future might be to address logo use in a written agreement up front when bringing in a new client/customer.
It depends on the context. For example, if you are reselling used Mizuno golf clubs on eBay, and you provide photographs of the actual clubs being sold -- which bear the manufacturer's trademark and logo -- that is permissible. Similarly, if you are writing a blog about Mizuno's new JPX clubs, you could show their logo as a referent to what you are talking about. On the other hand, if you simply decide you like the logo and want to use it as wallpaper for a website that has no connection to Mizuno, then, no, you can't do that.
I would suggest you discuss exactly what you want to do with a reputable local IP lawyer. It really depends what the intended use of the logo is, and it's a very fact-specific inquiry.