If I understand properly, your domain is example.com
Subdomain1 is site1.example.com and will be pointing at 192.100.101.102 at one hosting provider
Subdomain2 is site2.example.com and will be pointing at 201.291.58.23 at another hosting provider
You need to find out the IP address for each of the different subdomains. Wherever is hosting it will be able to give you this info. If the host is insisting that you have to change the nameservers to their nameservers they are wrong.
Question is, who is controlling the DNS services for example.com? You can't have different sets of nameservers for a domain and its subdomains. There is one set of nameservers per domain and that nameserver will be in charge of directing the subdomains to their respective IP address.
If you are controlling the DNS through WHM, you go to DNS Functions -> Edit a DNS Zone -> (select domain to edit) -> At bottom add a new entry
First column: The name of your new subdomain (put site1 if its going to be site1.example.com)
Second column: 1440 (should be there already)
Third column: A
Fourth column: enter IP address where subdomain will be hosted.
If you're not controlling the nameservers for your domain, you have to ask whoever is to do this for you. If the name is at eNom, you can use their DNS services to accomplish this instead.
Let me know what questions you have or what step you're stuck at or need clarification.