There is a mountain bike riding group in Saigon but I haven't joined any rides yet. From what I've heard they don't ride often during the rainy season to protect the single track.
Contact phattireventures.com in Da Lat for some of the best mtb trails in Vietnam. The company is owned by an American and offers a variety of other action oriented outdoor activities. They have excellent mtb guides. I think they still offer the 37 kilometer downhill trail from Da Lat to the coast. Da Lat is about a 6 hour car ride north of Saigon at 5,000 ft elevation. Phat Tire should have all the latest contacts for bikers in Saigon too. They only recommend one guy for maintenance. Trust him only and never trust the local bike shops.
I hope you're bringing your bike(s) with you. You won't find much in the way of support or parts here. Bring some extra tires and tubes with you. The only tubes I can find are of inferior quality. I had my son bring some Schwalbe tires and decent tubes. I'm glad I brought two sets of rims, each mounted with road tires or knobbies. I live in Saigon and bike almost every other day on these street. But it is dangerous as you say. And more so each year with the increased congestion and construction causing anxious motobikers to push harder to get home.
I kind of stepped into biking these streets gradually. At first I walked everywhere and discovered the one rule, there are no rules. Very true. Then I biked around with locals. They whizzed through the streets and I had to tell them to slow don't so I could learn without getting killed. Very strange but cars and truck must stay in the left lanes at all times, motorbikes and cycles in the right lane. Cars turn right from the left hand lane. Motorbikes and cycles turn left from the right lane. Bus can use either lane. DUH!!!! However, these rules apply only when a cop is at the intersection.
We have two advantages though. We are defensive drivers and they are quite agressive. They live by the unwritten law of "the bigger vehicle has the right of way, always." Pedestrians are of last concern. The second advantage is that police come down hard on them for hitting a foreigner. It's kind of an unwritten law that "don't hit a foreign pedestrian, cyclist or motorbiker.
I see a few road bikers around town but these roads can be tough on a rims and tires. Sewer grates are square and many are in disrepair, leaving ruts the width of a tire in the same direction as traffic. A round sewer grate would solve that problem, but Duh!! again. The best road biking option is to go to the countryside. I'm moving to the new southern part of town to avoid traffic and congestion. It's a difference of night and day. I can hop on the bike and be in the countryside within ten minutes. I've had my fill of old Saigon and it's pollution, trash, traffic and noise.
My motto is "Give me a bike and a pool hall and I can live anywhere."