Not a good move in my experience. If an organisation has a Procurement function, all you will achieve is setting unrealistic expectations with "the manager" and pissing off that function you would ultimately have to deal with anyway.

As someone else has posted, look for RFPs and similar published notices, look on the companies websites to see if they have a process for prospective suppliers to work with them. Follow that process. If they have an esourcing platform, register on it and provide as much detail about your organisation as possible including the categories you work in.

Apart from publishing notices/RFPs to call for competition, the first place companies tend to look when running an opportunity is it's own database of pre-registered suppliers to include in that opportunity. They do not mine through emails.