Most large CRM vendors utilize multiple IP addresses and/or servers to get around the fact that car dealers are considered one of the top 10 largest spamming* businesses by the email industry. If they've built things well then they know when one of their IPs is having issues. All it really takes is to check to see if some of the larger email services are sending messages back. Some will even tell you why they're blocking your messages. This part does take manual checking though. Ask them if they've done this yet. Also ask how many IP addresses they're using for your store and how many other dealers are tied to those too.
Whether you use your own domain or not is more a matter of convenience for the newer CRM systems out there. Newer CRM systems can mask your domain over their own, but that can sometimes look shady. Some of the older ones (especially ones originally built in the 90s) may actually use one of your own servers instead of their own and then you have to do the negotiating with the email services when (not if) you get blocked.
I don't know which CRM you're on, but CRMs originally coded prior to the Internet being a full blown fact of automotive retail typically do not have email in their core code. For example, there's a CRM
@Jeff Kershner was on, during his MileOne days, that required somewhere north of 10 clicks to add an attachment
Unfortunately there isn't a single CRM of any decent client-base-size that was built in this decade. There was one at one time...
*When we were building Dealer.com's CRM we put roughly 20 RFPs out to various email providers like Constant Contact and MailChimp. All of them basically gave us the left-handed turn down by giving us 5 billion hoops to jump through and some horrendous quotes. When we pressed them as to why prices were so high they told us that there are enough rotten-egg car dealers spamming to make the industry one they like to avoid in the kind of mass we represented. We were going to be bringing them around 20 million emails per month at that time, so they had to look at us as one big damaging bulk for them. I totally understand and appreciate their stance after we built a fresh email system to handle our own loads.