Can I use Weedle's "Speed Evolution" attack to search my deck for Beedrill and place it on Weedle?
No, you can't choose Beedrill from the deck. Beedrill states it evolves from Kakuna, so it is not a card that evolves from Weedle.
Source: Unleashed FAQ (2010-05-13), PUI Rules Team (2010-05-13)

Relevant cards

Note: All applicable cards may not be listed below.
HGSS2_32
HGSS2_69