Problems accessing a particular website

The difference isn't going to show when tracert succeeds on both machines.  Are there other sites that are acting the same way?  I have two different ISPs and I don't have any trouble with either of them.

Traceroute will continue to succeed even though I cannot actually pull up the website.  I did the traceroute to see if there would be a failure and there was not.  However, I opened up all of my web browsers when using the wireless connection and none of them could reach the website.  I have no problems with any other website.  
The part that I did not include is that both networks go through the exact same ISP, with IP addresses .85 and .89, so they are even on the same subnet and external gateway.

Tracert and web browsers use different protocols so that's not entirely surprising.  You probably need to use Wireshark https://www.wireshark.org/ or some other network analyzer to see what the differences are.

Have you taken a PC from one network and tried it on the other to rule out any other differences (domain/workgroup, IE settings, AV software etc)?

does the website OPC.org relate to the internal systems at all? (IE is that their own website or an external, unrelated company)?

It is an external website.  I have used my computer which is connected to both networks to reach the webpage.  
The website opc.org is not related to internal systems at all.

I have used my computer which is connected to both networks to reach the webpage.  

and does it work on both networks? (unless you mean it is connected to both at the same time, in which case that isn't a valid test)

I used both connections on my machine to try to reach that website.  I would turn off one connection and use the other connection to test.  So, at one time there was only one connection active.

so to confirm: it is a dual-homed machines which has a NIC on each network.

Please retest on a machine that is only on one network or the other and let us know the results as this machine could have any number of factors that invalidate the test (eg DNS cache, dynamic routes, ARP table entries)

Every computer, roughly 150, that are on the "wireless" network do not resolve to opc.org.  Every computer that is on the "wired" network does resolve to opc.org.

that is a very interesting distinction and seems suspicious. Did you do the test with a single PC tested on each network independently (not the dual-homed one though)?

do they use the same DNS servers and the same default gateway?
Do they get DHCP from the same server?

Most of the people using the wireless are students.  I have had complaints every single time the students try to access opc.org.  
DNS is handled for the wireless network publicly (8.8.8.8 and 8.8.4.4)
DNS is handled for the wired network internally with those servers pointing to 8.8.8.8 and 8.8.4.4

The two networks are two completely different cable modems, but those two modems do use the same external gateway.  

DHCP for the wired network is handled by one of my Windows servers.
DHCP for the wireless network is handled by one of my routers.  

These two networks do not interact at all internally.

Great. Thanks. How about the test with the same pc on each network?

Do you mean testing with two computers with identical specs on each network?

No. The 'same' computer.

My computer is set up via DHCP on both networks, so it would have the same settings as every other computer in the networks.

I don't agree. There are considerably more settings in a pc than DHCP can provide.
Have you done the test or not?

I am not exactly sure what test you are telling me to do.  You want me to take a machine from one network and put it in the other network?  I have done that multiple times and the machines always load opc.org on the one network and never load it on the other network.  When I say always and never I am speaking 100% of the time.  

What other settings are you asking me to change?

Never said about changing settings, and yes, that was exactly what i have been asking you to do, assuming you dont mean the multi-homed pc.
Thanks for the results.

Have you tried connecting to the wifi network by cable (spare port on the router maybe) to rule out the wifi bit?

I have computers that are connected to a switch coming off the router which is also the same piece of equipment that my access points operate off that have the exact same result.  I believe I have even connected directly to the router and the same problem exists at that level too.