I recently installed proxydroid app in my android 2.3.6 This application requires root access. Every time to use it, I need to make root access to my device.
Is there any proxy app which does not require any root access?
Simple answer: No.
No, as these app use a native binary installed onto the system to reroute data through the proxy. This is done at kernel level (actually it's done using the terminal commands, but it is done at kernel level), because of this root access is needed.
There are 0 apps that allow you to use a proxy without rooting first. Android 4.x natively supports proxies however.
No, proxy apps available on the playstore all need root (to my knowledge). But you can use the native proxy -- even with Android 2.1.
As Liam wrote in his answer, an explicit proxy requires kernel level access. But Android has a native proxy built in.
Before Android 4.x, its settings were simply hidden. So all you need is access to its settings page -- which is what some apps provide, as e.g. Proxy Settings does:
From the app's description:
Proxy Settings works with Android 2.1 and higher.