Graphics card damaged in mid 2011 27" iMac, is this a part I can find to replace?

Back up using Time Machine! If you have never backed up it's incredibly simple. Your computer has Time Machine built in, all you need is an external HD. I'd recommend the OWC Mercury Elite Pro. If you have never heard of Time Machine then please click Back up your Mac with Time Machine. When you get a new Mac you do the following:

When you buy a new Mac the easiest, quickest and most reliable method is to simply connect the old Mac's Time Machine External Hard Disk to the new Mac using a USB cable before you turn on the new Mac. When you start the new Mac for the first time it will startup Setup Assistant (SA), SA will get to a point where it asks if you are migrating from another computer, select Yes  and follow the on-screen prompts and it's done! It is THAT simple.

If you have already done SA on the new computer then simply connect the Time Machine External Hard Disk using a USB cable and launch Migration Assistant (MA) which is located in Applications - Utilities - MA. Then follow the on-screen prompts. 

The ONLY difference between MA and SA is MA will create a new user account that you need to log into.

Both SA and MA migrate applications (not 32 bit apps) , data files, settings, it does not migrate Mac OS itself. Some professional apps such as Adobe and Microsoft apps may require re-installing their apps as they will not allow them to be migrated. As long as you kept the security keys for installing their apps, it is not a big deal. 

After the migration is done, you leave external HD connected and let the new machine back up to the same External Hard Disk.

Original install parts have never been available anywhere (except to authorized service providers). You can try a do it yourself site such as ifixit or you can go to a service provider and see if they can do anything.

Do check on the cost - this is a 10 year old computer which is considered obsolete, so I would not spend a lot of money on it. Better to save that for a newer model.

The board is replaceable but you will enter the wild and woolly world of the used/pulled market because Apple won't have them. I've researched those exact models and they are always in high demand. If you find one, it will likely cost more than your computer would sell for on the used market were it fully working.

A lot of 6970 boards were replaced due to manufacturing faults so that reduced the pool of used, known-good boards. I've bought Mac parts from this place and they are sold out of both boards. Note the cost when they do have them:

https://www.powerbookmedic.com/Intel-iMac-27-AMD-Radeon-6970M-1GB-Video-Card-Mid-2011-p-132069.html

The 6770 is only slightly less painful at US$200, but they are out of them as well, and have been for months.

Frankly, if you want to continue using that specific model iMac, you could find a healthy one for the cost of one board. Old computer parts always cost more than old computers.

Sorry, you are asking the wrong person for this - ebay sellers have tried to scam me several times so I don't do business there at all any longer. I wouldn't trust anyone's claims unless I can verify and a "refurbished" card at that.....

You cannot, you will have to physically remove the internal HD and put it into an enclosure. Then connect it to the new Mac and use Setup Assistant to migrate your data.

Spending $200 on a machine valued at about $300 simply does not make sense. IMHO, I'd move on a look at replacing it.

If the drive is still alive, take it out of the Mac, put it into an external enclosure, and plug it in to your current mac.

If it is a regular spinning drive in a desktop computer, it should be a 3.5".

And, as I said previously, you'd need to remove the hard drive.