We have a web service used by InfoPath forms that we would like to continue to use in Office 365.
The guidance we have received is to use Live IP to connect the Azure hosted web service to our on-prem database.
Is this a common practice?
Are there better options? (assuming that moving the data to Azure is not possible at this time)
Use a site to site VPN or Point to Site VPN depending on how you can code your application.
This will insure you data does dont use the public internet, the service you are using can then use local ip addresses or DNS names if you have configured a DNS that is a available to it.
However you may not get the results you want to due to the delay from cloud to on-prem and back.
here you may wanna consider express route.
i just wanted to clarify after re-reading my response.
You can give cloud Apps and Cloud services Vnet access, you can then use a site to site vpn on that vnet to your on-prem data.