Your question is about DNS, if your MX records are specified on your current website host – then yes, moving your DNS nameservers to a new website hosting company would require you to enter your MX records. This is something a website hosting company assist you with.
Moving your DNS records to a new location does require time for the new information to propagate, and you may have a delay in receiving email. How to keep that delay as short as possible is something I would again discuss with your possible new hosting company.