In D&D 5e Adventurers League, can the siege weapons as seen in the DMG be purchased?
No, you cannot buy siege engines in AL (currently).
(This writing is as of the Adventurers League Player's Guide v8.2.)
Part 1 of the ALPG includes equipment at character creation in Step 5; this refers you to items "the resources listed in Step 1 & 2," none of which are the DMG (or contain siege engines like those in the DMG).
Part 2 of the ALPG describes rewards and equipment separately. Equipment can only be bought/sold between sessions as in character creation: from the books that don't include siege engines. During play equipment can only be bought as laid out in individual adventures; I've not seen any that allow the purchase of siege equipment yet. (Though I've not run all of the AL adventures: if some enterprising reader knows of one and can point me to it I'll gladly delete this answer and bounty theirs whatever votes this had!)
Rewards include treasure (check)points, which can buy you magic items from tables A-I in the DMG, subject to tier limitations. We're a little closer to siege equipment, but not there, sadly.
Unless you think a horn of blasting counts. It worked for Joshua, after all.