Need Someone to Vectorize our logo

We are a rescue Zoo located in Clayton, Delaware.  We have a newly designed logo, but need it in high resolution for multiple purposes including website, large size truck magnet, t-shirt (screen printing), etc. Specifically the t-shirt printer has asked for the following: a high resolution vector file and can color separate it. Let them know it is to be used for screen printing and they should know what files you need. Also make sure that you get the PMS colors they used for the logo so I can match them.

I have no experience with this and attempted to try CorelDRAW to help, but my computer must not have enough memory to execute the processing because I get an error message every time.  I think the minimum resolution is 3,000 dpi. I have some files that a marketing company sent to me, but none of them were what I needed in hi-res.