Last night I was at a wedding, and when I stood up as the bride was passing by on the way to the Chuppa, a fellow in all seriousness told me "It is not proper and a lack of Tzniyus to stand up for the bride". I told him that I am certain that there is a valid source for doing so as we see that it is a Minhag Yisroel that people stand up when the bride passes by. What is the source?
See Halachically Speaking (Volume 4, Issue 12, Page 8) where the author brings that many poskim [see footnote 108 for names] actually say to stand the entire Chuppah. (One reason given is because the Chosson is doing a Mitzvah, so we stand in his honor). Common custom however, is not like that.
He then goes on to say:
It is customary to stand when the chosson and kallah walk down the aisle. Some say this is out of respect for the chosson and kallah.
He brings Knesses Hagedolah E.H. 62[:2] as a source for that last reason [though the Knesses Hagedolah is referring to standing for the Chuppah Berachos].
So there you have it: a source.
I don't have a source for the custom, but I think it's irrelevant. The onus is upon him to show a source that it isn't tznius. Standing for a woman in general certainly isn't a violation of tznius - in fact the Gemara says one is obligated to stand up for the wife of a talmid chacham (Shevuos 30b).
I have heard it comes from the mishna (Bikurim) that says people in Jerusalem would stand and greet people bringing bikurim, on which the g'mara (Kidushin) says chaviva mitzva b'sha'tah, a mitzva at its right time is beloved [and we stand up for the one doing it].
However, I'll have to find the source that connects that to a bride (and groom).
See also the comments on this answer.
The Jewish Press of the week of January 9, 2016, brings a small piece by Rabbi Soleveitchik saying the tzibbur stands when the chassan and kallah walk down the aisle to honor the parents that are bringing their children to do the mitzvah of kiddushin. I think he also mentions the mishnah in bikkurim as a source to why we stand to honor the parents in this case
Quotes from two of Rabbi Leib Tropper's Shoel Umaishiv emails:
Dec 24, '15:
Rav Moshe [Feinstein], zt'l took the position that the choson only earns the status of 'Choson' after the chupah. Therefore if after the Choson's Tish people would daven mincha he would also say 'Tachnun'. He would not stand up for the Choson or the Kalah as they walk to the Chupah since they do not have the status of Choson & Kalah until after the chupah.
Nov 19, '12:
Maran Harav Moshe [Feinstein], zt'l did not stand up for the choson & kallah as they walked to the chupah. Rav Moshe's position is that they are not a Choson or Kallah until after the chupah.
He also did not exempt the choson from tachnun at mincha before the Chupah as he is not considered a choson until after The chupah.
There are other poskim who concur with this psak.