Can I buy (fill in the blank)?
If the gear is legally available (has no Availability rating), you can buy it from your local Stuffer Shack, Weapons World, or Kong-Walmart at any time, provided you have the available nuyen. Gear with an Availability rating isn’t as readily available, and as such you need to go looking for it, or have your contacts go looking for it.
If you go looking for a piece of gear, use the standard Availability Test rules (see p. 418, SR5) and mark off the appropriate amount of time on your Missions Calendar. Note that you can only search for one item at a time, as this represents you physically going out and looking for the item, spending time making phone calls, talking with people, buying them drinks, etc.
If you wish to have contacts purchase additional items during a single downtime, use the standard rules for determining what items they can find, and how long it takes (see p. 418, SR5). Since you won’t always know the skills and attributes of all your contacts, all contacts instead use their Loyalty instead of Charisma attribute and Connection in place of their Negotiation skill for the Availability test.
You cannot buy used gear.
So, does this mean character-bought contacts are effectively useless for buying gear with an Availability of 8+? The section on Black Market Goods describes it as an opposed test of Negotiation (which is Connection) + Charisma (which is Loyalty) versus the item's availability rating. The maximum amount of dice a starting contact can have is 7 (6 Loyalty and 1 Connection, 1 Loyalty and 6 Connection, or something in the middle). This means they can only buy 1 hit, while an Availability 8 item gets 2 hits.
Basically, unless you're a troll with the social grace of a rabid wolverine, it's going to be easier and cheaper to ignore your contacts and search for the item yourself...or go to another player's face character and bribe him to get the items for you. I could see an optimized face (20 dice is possible at character creation) swimming in Nuyen from purchasing items for others and charging a small premium.