I want to enter my Saison into a competition. It will be my first.
I have it carbonated to 3.4. Do I need to buy special bottles to withstand this much pressure? I don’t want any dangerous accidents.
I don’t want to buy thicker bottles and a new capper if I don’t have to.
You need thicker bottles. IME, I have lost some bottles when carbonated at 3.0 v/v. I have not had to get a new capper for thicker bottles but I suppose that depends on the bottles you buy.
You are better off using thicker bottles, especially if you are shipping them where the bottles may be handled roughly. Some of the commercial longnecks will hold this pressure but I try to avoid putting higher carbonated beers into standard bottles.
There are some good options in the 12oz/375ml range. The bottle shape used by Almanac and several other American sour brewers can withstand a lot of pressure. The stubby Belgian 375ml bottles are good as well. Some Belgian brewers also use longneck bottles that have a smoother curve from the body to the neck that hold pressure well. Leffe uses these bottles. All will accept a 26mm cap. If you’re just bottling a handful of bottles I’d think about getting some commercial bottles and enjoying the contents to reuse the bottles.
Boulevard’s ‘Tank 7’ would be another good commercial option for thicker 12 oz bottles that’s usually not too hard to find. You get very tasty beer with the bottle as a bonus. 
Thanks! It’s always great to get bottles with free beer in them.
I have had good success bottling from the keg with a beergun between 3 and 4 vol CO2. Granted some is most likely lost during the bottling process, but I have achieved close without bottle breakage even in the crappy homebrew shop bottles.
I do agree with others here that rough handling can play a role in breakage, but it can be done. Also helps if bottles can remain cold for most of the storage time prior to being judged.