This came up in our skirmish today.

Can a E-web w/ Vader's Finest, use the first ability listed, and move 2 after performing an attack?

No, because of the wording on e-web. You cannot voluntarily exit your space (which is what spending movement points is) if you attack.

The E-Web Engineer with Vader's Finest could use the special action to perform an attack and gain the 2 movement points, but it cannot spend the movement points due to Tripod .

It could even spend the other action for Move action to gain more movement points, but then be unable to spend them due to having attacked during the activation.

(However, if there are no valid targets for the attack from Vader's Finest, then the E-Web would not have performed an attack, and could spend the 2 movement points. But a normal Move action would be better. The case could still happen with some command card where the intended target moved away from line of sight. But you probably would not put Vader's Finest on an E-Web Engineer in the first place.)

If you have Imperial Industries attached and can use Executive Order or some Agenda cards, that lets you attack with the E-Web outside his activation, you could get the effect.

(One surge ability in the Imperial Industry reward card is surge: after the attack resolves, gain 2 movement points.)

The E-Web Engineer would need to get that surge during the attack, which is fairly probable but not a given. (And the attack needs to be outside of his activation to avoid Tripod triggering to be able to spend them.)

If there's ever two rules (coming from other cards, not the core rules), one saying you can do something, and one saying you can't. The one saying you can't usually takes precedence.