Whenever you want to talk directly with the Oracle server, the proprietary protocol called SQL*Net is needed, and that comes with "Oracle Client" installation.
If u really cannot install Oracle client, eg, on so many machines (remotely or not) that need to talk to the Oracle server, it is possible, then you need 3-tier, plus converting the VB.net to a browser - based application.
First tier is your client - which for our case, we can use a browser, which is installed in every computer nowadays. That will talk HTTP traffic with a tier 2 server (or called it our homebrewed application server). And you can used this tool to help out the conversion:
Tier two: this is a Apache server with Oracle client installed, plus other component that can convert the HTTP traffic coming via Apache Server (or any HTTP server) to that of VB.net client. Just one tier 2 machine is needed.
Tier 3 is your database server.