I've looked on the websites and, other than just going in and asking each and every local synagogue, is there some easy way to know if an orthodox synagogue is Modern or Traditional?
The question mistakenly assumes that every Orthodox synagogue is either Modern Orthodox or not. This is not the case.
The MO and charedim mix to a large extent. There are shuls with a charedi rabbi but a mainly MO congregation, and there are many shuls with a mix of charedi and MO congregants.
Many people are in between MO and charedi. They might describe themselves as "frum but not yeshivish" or "centrist Orthodox" or just "Orthodox" or "frum." The men will often wear black kippas, typically velvet, but not black hats, and will wear colored as well as white shirts. Other people may "look" charedi but have a more MO hashkafa or halachic orientation, and vice versa.
Accordingly, I'll answer the question by noting separately how you can determine whether the rabbi and congregation are MO or charedi.
To determine whether the rabbi is MO or charedi, look at his bio. Did he go to YU, and does he belong to the RCA? Then he is probably Centrist or Modern Orthodox. Did he go to YCT? Then he is Modern and/or Open Orthodox. Did he go to a charedi yeshivah? Then he's charedi.
To determine whether the congregants are MO or charedi, see how they dress. The more knitted and non-black kippas, the fewer beards and especially long beards, the fewer black hats, the more MO the shul is.
A high percentage of black hats is a sign that the synagogue is more haredi orthodox than modern orthodox.
A prayer for the state of Israel is a sign that the synagogue is more modern orthodox.
You can also look at the tone of the parsha sheets and listen for the tone in the Rabbi's speeches.
If there are some people who regularly drive to the synagogue on Shabbos, it is almost certainly not a haredi orthodox synagogue.
In America, using an Israeli/Sephardi transliteration (Shacharit as opposed to Shacharis) often indicates that the synagogue is more Modern. The size of the mechitzah is also a clear indicator; the larger and more opaque it is, the more traditional the synagogue. Modern congregations usually sing more of the prayers, so their Shabbat morning prayers are usually longer.
Here's an interesting contrast to use an example:
One shul that I attended had the women daven on the 2nd floor. Not a balcony - but I mean really the 2nd floor. There was a hole in the floor of the women's section so that they could hear the davening. (A fence surrounded the hole so no one accidentally fell through.) This was very Orthodox.
In contrast, Lincoln Square synagogue has "stadium style" seating using concentric circles as seating rows. There was a mechitzah, but men and women could clearly see each other (and, did - hence the nickname "Wink and Stare".) This place was Modern Orthodox.