It takes a while, but stick to making Duos with someone for a few hours and you'll be getting that armor within a couple weeks. Each Duo should take you about 20-30 minutes to clear, and you get 2 Marks per run.
Personally, i haven't found a better method, but i'm also open to suggestions.
Also, every now and then, you get some from the Vault, so you better not be skipping any visits. --- GFWL: GrifoPT WFC: Griffon
Duos shouldn't take you longer than 20 min. You can also queue up for T1 Hard Alerts. If you do ALL the duos and ALL the Hard Alerts you can actually get close to getting about 1 tier 1 piece of armor every 1-2 days depending on which piece it is.
There are plenty of ways to get your marks of Triumphs, doing your duos is a must, especially the Hive and Omac since you get an extra mark from those 2. Also Challenges give you 1 mark of T and Toyman Misisons, thats another 3 Marks for 20min of work, and don't forget your vault. You can easily do 20+ marks of Triumph a day, more so if you do alerts, so it all depends on how bad you want them.