Here's the breakdown:
Step 1: Pair Arbor Elf (Equipped with Bracers) to Galvanic Alchemist when the alchemist enters the battlefield
Step 2: Tap Breeding Pool for 1 blue mana
Step 3: Tap Arbor Elf for its activated ability, Bracers copies the ability, Giving you two "Untap target forest" abilities on the stack, as you said. You let the first resolve, then respond to the second one by tapping your now-untapped breeding pool for one mana. Then, after the second ability resolves, you tap the Pool again for one more mana
Step 4: Now, with three mana in your mana pool, you activate the ability Galvanic Alchemist granted to Arbor Elf. Because Soulbond gives Arbor Elf the activated ability (each of those creatures has "2U: untap this creature,") you treat the ability as though it were printed on the Arbor Elf card, therefore Illusionist's Bracers copies the abilities.
Step 5: You let the first "untap arbor elf" ability resolve, then you respond to the second by tapping your now-untapped arbor elf.
Substep 5: Illusionist's bracers copies the activated ability and so you repeat step 3, except instead of tapping the pool a second time, you leave it untapped, providing you with a net of one mana.
Step 5 cont: the second "untap arbor elf" ability resolves, now you are left with an untapped arbor elf, an untapped Breeding pool, and one blue or green mana in your mana pool. Rinse and repeat for infinite mana.
I know this is an over-explanation, but I ran through it this way so I could keep it straight in my head. I think what answers your question most concisely is that the soulbond ability gives arbor elf the untap ability, which is then able to be copied by the bracers.
Hope this explanation was helpful!
