Here are results from an online IP lookup tool for the www.arcgis.com domain:
Source: whois.arin.net
IP Address: 23.21.110.28
Name: AMAZON-EC2-USEAST-10
Handle: NET-23-20-0-0-1
Registration Date: 9/19/11
Range: 23.20.0.0-23.23.255.255
Org: Amazon.com, Inc.
Org Handle: AMAZO-4
Address: Amazon Web Services, Inc.
P.O. Box 81226
City: Seattle
State/Province: WA
Postal Code: 98108-1226
Country: UNITED STATES
Name Servers:
Source: whois.arin.net
IP Address: 54.197.254.10 (United States)
Name: AMAZO-ZIAD7
Handle: NET-54-196-0-0-1
Registration Date: 11/11/13
Range: 54.196.0.0-54.197.255.255
Org: Amazon.com, Inc.
Org Handle: AMAZO-4
Address: Amazon Web Services, Inc.
P.O. Box 81226
City: Seattle
State/Province: WA
Postal Code: 98108-1226
Country: UNITED STATES
Name Servers:
Source: whois.arin.net
IP Address: 54.243.30.195 (United States)
Name: AMAZO-ZIAD1
Handle: NET-54-242-0-0-1
Registration Date: 11/9/12
Range: 54.242.0.0-54.243.255.255
Org: Amazon.com, Inc.
Org Handle: AMAZO-4
Address: Amazon Web Services, Inc.
P.O. Box 81226
City: Seattle
State/Province: WA
Postal Code: 98108-1226
Country: UNITED STATES
Then you'll need to find the IP ranges for the list of domain names that Esri provided you with? Can you share those please?