Are there any groups or organizations that help with financial assistance for dental operations?

There are many groups out there, but you'll have to narrow down what's available to you in your location.

For example, Smiles for Freedom* has seven events planned this year at various locations around the US. Aspen Dental has their annual Health Mouth Movement* that provides free dental care for veterans.

Everyone for Veterans* connects veterans with local dentists that volunteer their time and services to provide dental care. The eligibility is Honorable discharge after a 2 year enlistment, can't be 100% disabled (you'd get free dental through the VA than), and you are experiencing "financial hardship" related to your dental care.

Additionally, check with your state's Veterans Affairs office to see if there are state specific programs that you might be eligible for. For example, Florida recently enacted the Veterans Dental Care Grant Program for free dental care to veterans that aren't VA eligible (was just passed at the beginning of July, so it's still in the works getting the money out).

You can turn up many, many more that may or may not fit your circumstances by just doing an internet search on phrases like "dental care for veterans", "dental assistance veterans ________" (fill in your state), etc.
------------------------------
* Smiles For Freedom - https://www.smilesforfreedom.org/
* AspenDental - https://www.aspendental.com/about/healthy-mouth-movement/
* Everyone for Veterans - https://www.everyoneforveterans.org/applynow.html

What kind of assistance, specifically? Assistance to offset the cost? Pay for it completely? You could try reaching out to a VSO, but what they could possibly assist with is won't be much (it's based on their by-laws) and the amount would be expected to be paid back.