Bootable usb for use on an older MAC

The iMac should be able to run macOS 10.11 so 10.10 or 10.11 will be easier & better options since I've sen another user mention that macOS 10.9 may no longer be available through the App Store (not sure if user had previously "purchased" it).

You will need access to another Mac from 2008 to 2015 in order to create a bootable macOS 10.10 or 10.11 USB installer:

Create a bootable installer for macOS - Apple Support

I don't know if you can install either one if the iMac is still running only on the system firmware from macOS 10.5, but may be worth a try.

PS: Older discussion thread:

What is the latest OS for imac 7.1 - Apple Community

says

(in 2014) "The iMac7,1 can go as high as Mavericks (10.9.5)"

NB: suggests updating gradually ie: to Snow Leopard (10.6.8) first.

Some downloads are available from: Apple - Support - Downloads

In this discussion thread:

How can I get mac os x 10.6 (snow leopard… - Apple Community

Neil says:

"check nearby or online stores which sell secondhand Apple products".

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I see SnowLeopard is listed here: Apple - Support - Downloads

(Haven't tried to download it though)

No ideas about the installer.

Wondering if the upgrade you want is compatible with the particular iMac model.

This article (link below) lists the newest compatible OS version that can be installed on various models (by year of manufacture):

Identify your iMac model

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Apple Support has an article online (link below) with links to older MacOS versions:

How to download macOS - Apple Support

NB" At the bottom of the article:

says:

"If the macOS that you want isn't compatible with your Mac or the currently installed macOS, the installer will let you know, and the App Store might prevent you from downloading it.

For example, it might dim the Get button, say that the macOS is not compatible with this device, or say that the requested version of macOS is not available".