adding a wifi router

But without knowing the Network Topology it's impossible to say one way or the other.

However what happens if you remove the Router and restart the entire system again?

Col

if you added a "router" [doesn't matter if it's WiFi or not], you've created a separate network for those hosts behind that router, on a separate subnet. Shouldn't matter to the network you added the router to whether you left DHCP on or not. It can't see the DHCP server because the router's DHCP service is on a separate subnet.

The only time this would become an issue is: 1. the WiFi router isn't authorized to be on the network. Then your guilty of installing a rouge device. Those who are the boss, aren't going to like it.
2. if hosts behind the additional "WiFi" router need access to the other network for services on the other network. The "other" network's router needs to know about the WiFi router's network and vice versa to route traffic between the two routers. Further, one needs to create firewall rules to allow that traffic. If the "other" network's router doesn't know about the WiFi router's network, then the "other" network's router will simply drop the frames.

Always look at the physical connections first. Most routers today are a router, with a wifi access point and switch. Are you plugged into the existing network via one of the "switch(LAN)" ports or the "internet (WAN)" port? If plugged into the Lan side, that could be whats causing the problem.

I like Col suggestion in the post above. If its not a physical connection issue, that is the next step I would take. At the very least you are ruling out and potentially determining what device is "not" causing the issue.

Also I would wonder the validity of the outside agent if he is pointing to the wifi router as the culprit merely because its a "new" or "newest" device on the network. It IS typical to look at newest devices and troubleshoot to see if it is a cause. Without actually troubleshooting, what is the reason he gave for the wifi router being the problem?

Your WiFi Router quasi Access point with DHCP is probably handing out wrong information to hosts via DHCP. It's not that the whole system fails, it's that clients have wrong information such as gateway and DNS that is essential for hosts to have to find resources on the network.

Turn off DHCP on your WiFi Router quasi Access Point. Go buy an Access Point, not a router, if you want to provide WiFi services to the network.

The outside network guy is right.

Why would it work for months without an issue? DHCP TTL. Network administrators set long TTLs for DHCP to reduce network traffic. Remember that DHCP TTLs of default 7 days actually have clients requesting renewals at half that time. If you have a lot of clients, having short TTLs can create a LOT of traffic on the network every 3 days, as clients continue to request renewals of their DHCP addresses [plus DHCP options] every 3 days. A long TTL means that clients can go months without requesting a DHCP renewal, but once they do and your WiFi router / quasi Access point with DHCP answers DHCP requests, you've just given clients an address [and options] from that DHCP server. Since default settings on DHCP on your WiFi router provides clients with default gateway information of your WiFi router/quasi Access point, and it's not a gateway, clients can't reach the internet. Just about every service that requires to leave the internal network won't work as the WiFi router/quasi Access Point being told to clients as the default gateway isn't a gateway.

If you run Directory services, matters just got worse, as the WiFi router/quasi Access Point also provides DNS information to clients. Since there is no DNS information obtained by the WiFi router/quasi access point's WAN port, [that's how these routers work] clients don't get DNS server information to resolve name lookups to IP addresses, such as Domain Controllers.