Dear Sirs,

This is a bit old fashioned and is nowadays more associated with a style used only by non-native English speakers. It's appropriate to use this if you are writing a letter using an old fashioned style, but if this is a modern day email to a department you do not have an existing relationship with, you can start the email with "To whom it may concern," or simply "Hello,".

We are looking for a new supplier for different kinds of medical supplies for our students, such as phonendoscopes, stethoscopes, neurological hammers and some other diagnostic sets.

We investigatedtook a look through the online catalog on your web-site properly and would be pleased if you could send us a complete catalogue with detailed descriptions and prices as well as delivery and payment terms and conditions.

Web-site is non-standard, so it's better to use the more standard spelling in a formal letter.
Catalog and catalogue are both acceptable spellings, but since you used catalog first you should be consistent.