If you're looking for authentic collection level memorabilia (match jerseys) from a brick and mortar store, your best bet is to go to a team's official store. Although CGCampbell is correct in that you'll be able to find some type of jersey for a local sports team at any given sporting goods store, it is unlikely to be an authentic jersey, even more so for a smaller market US league such as MLS. Always remember, "licensed" just means MLS accepted money to allow a company to use their logo, it says nothing about the quality of the product. Also, you will never see significant discounts on authentic jerseys, that's just the way these premium goods are marketed.
For example, I just did a search on Dick's Sporting Goods' website for both LA Galaxy and SJ Earthquakes jerseys and found they only sell "secondary" "replica" jerseys. "Secondary" meaning they only have the away game colors and "replica" meaning it is not the same quality as "authentic" match jerseys. Team stores will have both home and away colors as well as vintage and other special seasonal jerseys all at match level quality.
That being said, which team store should you go to? Out of your three specified locales only Los Angeles and San Francisco have MLS team stadiums within driving distance.
The LA Galaxy play at the StubHub Center in Carson which is ~15mi / 24km from both LAX Airport as well as Downtown Los Angeles. Open Tue-Sat (and game days) 10am-5pm. LAG team store webpage
The San Jose Earthquake play at the Avaya Center in San Jose which is ~45mi / 70km from Downtown SF and ~35 mi / 55 km from SFO Airport. Open Mon-Fri 10am - 3pm and starting 2 hrs before games. SJE team store webpage.
Note 2018 will be LAFC's inaugural season and Banc of California Stadium will presumably have a team store, at the time of this writing I could not confirm its existence. Also, the first home game will not be until April 29, 2018 so it would also be unknown whether it would open when OP will be visiting. I will update when confirmation is available.
Having lived in both the SF and LA areas for years, I can tell you the traffic to both locations can be horrific if you are travelling at the wrong time. I would say the most convenient option would be to go to the Galaxy team store on Saturday as soon as it opens. If that isn't possible, my next suggestion would be to attend a game, the 2018 MLS season starts early March and playoffs won't be until November. At least then the commute time isn't entirely lost for just a jersey.
Regarding selection, from the LA Galaxy webpage:
In addition to LA Galaxy apparel, the Team Store carries official team merchandise for the U.S. Men's and Women's National Team, plus merchandise from other countries and clubs.
It would still be unusual for a team store to carry jerseys for other teams in the same league. Therefore I would like to suggest another option. Contact any of your hotels you have reservations for and notify them you will be receiving a package. Then purchase the jersey through the MLS online store and have it shipped to the hotel before your arrival. It should be quite a bit cheaper than shipping it to the UK. The online stock will pretty much guarantee you get the team, color and size of your choice without the hassle of taking a side excursion out of your valuable vacation time.
Lastly, having sponsors as the primary logo on a jersey is just stupid. The only reasonable kits left in MLS are the NY Red Bulls, and maybe Minnesota. Some are just mind boggling, I'm looking at you Portland, with a big fat "Alaska" across your chest.