The closest thing that you will come to an "Organized" plublically available NFA shoot, nowadays in or just outside of Austin is the monthly Texas Tactical Carbine Match.
While technically not an organized NFA shoot, it is probably the only time and place that you could pretty much guarantee to meet up with an Austin NFA owner or two if you wanted to shoot a SBR, Suppressor, or maybe a Machinegun. Over the years, I have probably let a dozen+ different folks put rounds downrange through my guns after the shoot was over and I know that others have done the same.
I have lived in Austin for 6 years and to my knowledge there has only been one planned Assault Rifle / NFA type event which was hosted out at the old Tony N 183 Range approximately 4 years ago. By "planned" I mean, advertised on this site (among others), appliances and reactive targets were brought out, range was rented out, BBQ and there were probably 35+ people in attendance with about 6 or 7 of us with NFA firearms. If I remember correctly it was a member of this site who put the shoot together.
Over the years, I have bumped into a handful of other Austinites that own NFA (not including dealers) at the Austin gun show, Hill Country Range, and at Texas Tactical at either the former Blanco or now Hays County range. All in all I have maybe met 10 other machinegun owners and an additional dozen more folks that had SBRs and Suppressors.
As of right now there are only two places that allow unrestricted full auto in the Austin Area (that I know of) and that is the private Texas Tactical Range and Hill Country Rifle Range. Unfortunately, since one range is private and one is run by AKA "Crazy Richard" combined with a relatively small number of private citizen NFA owners in the area, I think the odds on getting a public NFA/MG shoot together in Austin in pretty slim.
I could also potentially see a small private NFA shoot being put together at or by Texas Tactical if they were so inclined as most NFA owners in the area seem to shoot at their range. However, from my perspective it wouldnt make sense financially for Mke to host a NFA shoot for folks who are already existing members of the range.
The biggest NFA event in the central Texas area used to be Keiths shoot out in Leona (between Austin and Houston). The last public MG shoot he hosted two years ago drew over 100+ folks from all over Texas with everything from a line of Beltfed 50s to suppressed 22s and everything in between. There also used to be a Quiet Line time, where the range was hot but everything had to be suppressed. Unfortuantely, this shoot is now by invitation only as it quickly outgrew Keiths backyard.
There has also been a annual shoot in Waco at Tiger Valley the past couple years which is billed as a Machinegun/NFA Shoot. I have not been able to attend since its inception, however information can be had at the link below.
www.thebannedofbrothers.com/index.htmSorry to be long winded but in my opinion your best places to see and publically shoot NFA in and around the Austin area are.
1. The shoot at Tiger Valley (Pros - Most Amount of NFA Cons- 1 time a year)
2. Go to a Texas Tactical Carbine Match (Pros - Monthly Matches Cons- small NFA selection)
3. Join the Texas Tactical Range and if you shoot out there on a regular basis you are bound to bump into afew other NFA owners. (Pros - Great Range you should join anyway Cons - Never really know when to go to bump into somebody who has a can or MG)
I would also second Chuck Tooke for purchasing NFA in Austin. I have been using him for going on 4+ years now every since Just Guns went out of business and have never had a problem and an extremely friendly/helpful guy at that. However, I have also heard good things about Steve @Sreadfire and David @ GunStore.
James
Austin, Texas
Edited to add: I have never seen Steve advertise hosting a public NFA shoot in the Austin area. However, it may be that he puts together informal NFA shoots on occasion that are outside the purvey of public knowledge/open invitation.