Earlier quoted context omitted.
Some Thinkpads that are not supported provide bootable isos that you can use to update your bios. It is my understanding that it also updates the firmware (but it doesn't explicitly say). Here is an E model that has a bootable iso: https://pcsupport.lenovo.com/us/en/products/laptops-and-netb...
Yes, many vendors for many models do the same. Even if they don’t you could install it via Windows. “Support” is more than just the ability to install an update. It’s to do with whether an issue affecting Linux is being addressed. In the case of the E and L series Thinkpads there are many shared components with the T series so your chances are much better.
Some enterprise PCs only have Windows, DOS and RHEL, SUSE options. There's no reason to use Windows when you can update BIOS and firmware via CentOS Live image with RHEL binaries.