You CAN get tier pieces from your vault.
Every week at reset you are eligible to choose ONE PIECE of gear from The Great Vault based on your accomplishments from the previous week. Completing more milestones does not get you more gear, but expands your choices for the ONE PIECE of gear that you can claim.
You will also receive a keystone based on the highest timed run you completed the week before, or one less than your highest untimed run. If your highest key was a timed +15 or an untimed +16 last week, you're going to get a +15 key in your vault this week.
Remember that you only get to choose one piece of gear from The Great Vault no matter how many activities you do.
There are three paths to expand your gear choices: mythic plus, raiding, and world.
To reduce confusion, let's just assume that you only play mythic plus and don't raid or do world content.
To get the most choices at gear by doing only mythic plus, you must complete 8 mythic keystone runs a week. Completing 8 runs a week will present you with three pieces of gear to choose from at reset.
That's the short version - to be able to choose from three pieces of loot from the Great Vault, you gotta do 8 mythic keystone runs. Now let's do the long version where we talk about how keystone level affects the item level of gear you can choose from.
wait, how many gear choices do I get if I do X number of keystone runs?
If you do at least one, but less than 4 keystone runs: you get one piece of loot to choose from in The Great Vault. Its item level will be based on the highest level keystone run you completed. (Look at the table below to see what the item level will be.)
If you do between 4 and 7 keystone runs: you get two pieces of loot to choose from in The Great Vault. The first item's level will be based on the highest level keystone run you completed. The second item's level will be based on the FOURTH HIGHEST keystone run you completed.
If you do 8 or more keystone runs: you get three pieces of loot to choose from in The Great Vault. The first item's level will be based on the highest level keystone run you completed. The second item's level will be based on the FOURTH HIGHEST keystone run you completed. The third item's level will be based on the EIGHTH HIGHEST keystone run you completed.
tl;dr
One thing that I want to make clear now that you've read this: it doesn't matter in which order you complete your keystone dungeons, but it does matter how many keys you do. The game makes a list of all the keystone runs you completed and sorts them from highest level to lowest level, then selects the first, fourth, and eighth highest completed keystone levels. This means you can continue to do keystone runs past the 8 required for your weekly milestones to give you more options at higher level gear. You can NEVER LOWER the item level of the gear you're going to get since the list is sorted from highest to lowest.
Example: You've been busy this week! You completed ten total keystone dungeons! Wow! The game has built a list of all the keystone levels you completed and sorted them from highest to lowest to determine what you're going to get for gear choices. Here's an example list:
+15, +15, +15, +15, +13, +12, +10, +10, +10, +9
The bolded numbers are the first, fourth, and eighth highest runs you completed. The item level of the gear you'll be able to choose from will be listed in The Great Vault's UI.
Let's say you want to raise the item level of one of your choices of gear, so you go out and complete some +12 runs. Using the example above, the +12s are injected into the list and cause your eighth highest run to become a +12. Here's what the new list would look like:
+15, +15, +15, +15, +13, +12, +12, +12, +10, +10, +10, +10, +10, +9