Frankly we do not know this. It's been claimed, but never confirmed. Moreover, the sources from the time period actually contradict the claim. Vines was serving well throughout the first tour as I documented here: http://tt.tennis-warehouse.com/showt...75#post6401875
That list shows conclusively that if Vines had any serious shoulder issues preventing him from serving effectively, and forcing him to serve sidearm, such a physical issue must have been brief. There are simply too many dates documented on which he was serving aces and therefore must have been serving over-arm (for lack of a better term).
And if we're talking about something brief, then by definition it is not serious. It's on the level of many such physical problems that all the players on these tours faced at one point or another, for brief periods (such as illnesses, or blisters, or muscle strains, etc.)
I have not found any source from the time period saying that Vines served underarm. Moreover, historians like Ray Bowers who have manifestly studied these tours more than I have, also do not mention it: http://www.tennisserver.com/lines/lines_05_11_22.html
All we have is a claim, I take it, that was made many years later. I don't think it's a question of honesty but there are instances in which people incorporate into their memory events that did not actually happen; or their recollection of when actual events occurred, and what persons were involved, is faulty. Those instances are not that uncommon.
What I object to primarily is this, if you say that Vines was serving underarm during some of the matches, it gives the impression that he was dealing with a debilitating injury during these tours; and the vague reference to "some matches" can be taken by some readers to mean that he served underarm occasionally, throughout the tour; or that he served underarm during some period of time, which for some people can mean quite a significant period of time. 10 or 20 matches can count as "some matches" in a series of 40 matches. And it's just not true that Vines was crippled for 10 or 20 matches. At most he had a brief injury. That should be specified.
And all the players in these tours were dealing with brief injuries.