I sent a message to a shop, but I only filled in the email/message portion. I’m not sure if the message actually went through because I didn’t fill out the phone number.
It’s hard to give a definite answer as there are a number of variables that will control the outcome. However; in most cases, yes, the message will still send to the store owner if you leave out your phone number.
A general email response time is 24-48 business though this varies from business to business. If you haven’t heard back from the business owner then I would recommend searching the footer of their website for their social media pages to see if you can connect with them through other means.