Unseen servants have a speed of 15 feet. Can they take the run full round action or otherwise keep up with their master moving at a normal base land speed? Can they do this while pulling a cart (probably in concert with more unseen servants)?
There is a speed figure for an Unseen Servant, along with a list of available actions (emphasis mine).
It can run and fetch things, open unstuck doors, and hold chairs, as well as clean and mend. The servant can perform only one activity at a time, but it repeats the same activity over and over again if told to do so as long as you remain within range. It can open only normal doors, drawers, lids, and the like. It has an effective Strength score of 2 (so it can lift 20 pounds or drag 100 pounds). It can trigger traps and such, but it can exert only 20 pounds of force, which is not enough to activate certain pressure plates and other devices. It can’t perform any task that requires a skill check with a DC higher than 10 or that requires a check using a skill that can’t be used untrained. Its speed is 15 feet.
There is nothing that prohibits double move for an Unseen Servant. One can even argue that it can literally run, but even double move equals 30 feet per round and is sufficient to catch up.
As a side note, while the speed is not listed as fly speed, certain low-priority sources (Sage) speculate that Unseen Servant can be directed anywhere within spell range, provided there is a line of effect to the target location.
If the DM rules that an unseen servant follows the rules for creatures with Constitution as a nonability, an unseen servant can run indefinitely
An unseen servant isn't assigned Dexterity, Constitution, Intelligence, Wisdom, or Charisma scores, therefore it's not, by the Monster Manual's definition, a creature. But if it's creature enough for the DM to want creature-like rules for it, an unseen servant would have Constitution as a nonability (MM 312) which says that
A creature with no Constitution cannot tire and thus can run indefinitely without tiring (unless the creature’s description says it cannot run).
Note that if considered such a semi-creature, the unseen servant's specific movement rate overrules it Dexterity nonability, and it's specifically not an object but a force, overruling its Wisdom and Charisma nonabilities, but an unseen servant fails all Intelligence checks because of its Intelligence nonability.
Alternatively, in the Wizard's Workshop column "Sage Advice: Questions and Answers" from Dragon #301 (Nov. 2002), Skip Williams answers the following question:
Can an unseen servant run? Is it subject to encumbrance?
Though the spell description mentions "running and fetching," an unseen servant cannot use the run action. An unseen servant has a Strength score, and it is subject to the encumbrance rules. It moves at its full speed of 15 feet only when carrying a load of 6 pounds or less. If carrying 7 to 20 pounds its speed is 10 feet. It also move at a speed of 10 feet when dragging 7 to 100 pounds. Because it has a Strength of 2, it can life as much as 40 pounds (double its heavy load), and can move 5 feet each round (staggering) when doing so. Note that an unseen servant must move along the ground when dragging something, but it can move in three dimensions when carrying anything it can lift. (112)
Although your question's tagged [dnd-3.5e], this ruling comes from the Dungeons and Dragons, 3rd Edition era but still stands--as much as any Dragon magazine material does--, unless it contradicts the Dungeons and Dragons 3.5 rules, which, technically, it doesn't.
Thus Williams says an unseen servant can't run, but for this he provides no justification. It should be noted, though, that in Dungeons and Dragons, 3rd Edition a creature with Constitution as a nonability always failed Constitution ability checks (MM (2000) 10), which were (and still are in 3.5) required after taking the action run for rounds equal to the creature's Constitution score. Thus if a 3rd Edition unseen servant were permitted to run, it wouldn't be able to run for long. (It'd run for 1 round then fail the DC 10 Constituon check because of its Constitution Ø and then have to... um... rest, I guess.)
Williams's assessment of the unseen servant's carrying capacity appears correct despite edition changes.
I'm of the opinion that the nonabilities rules were vague enough during Dungeons and Dragons, 3rd Edition that it was just easier if an unseen servant couldn't run, and that with Dungeons and Dragons 3.5 these rules were clarified enough to permit an unseen servant to run, but ask the DM.
Further, I suggest acquiring a collar of perpetual attendance (2,000 gp; 0.25 lbs.) if in frequent need of unseen servants.