Is it possible for two Orthodox Jews to be married by anyone other than a rabbi?
Yes, but it's not recommended-
The "orchestrator" of the wedding makes sure nothing goes wrong.
The sages say that anyone who does not know the marriage laws well should not get involved with them (kiddushin 6a), as many mistakes can result.
פרש"י : בטיב גיטין. בהלכותיהן : לא יהא לו עסק עמהם. להיות דיין בדבר שמא יתיר איסור ערוה וזהו עיוות שאינו יכול לתקן
Rashi there: Shouldn't have any involvement, as he may come to allow a prohibited relationship
Although R' Yaakov Kaminetzky in Emes L'Yaakov reportedly (it is published מפי השמועה) allowed a grandparent or close relative to conduct the wedding
מותר לסבא או קרוב שכיבדוהו לסידור קדושין לקבלה אע"פ שאינו בקי בעניני קדושין שאם אירע איזה ספק בדין אפשר להתקשר למורה הוראה בקלות
this is usually understood in light of the Shvus Yaakov (3:121)
ובודאי לא בחינם הורגלו דורות הללו שאין מסדרין קדושין בלתי התרת הרב כי יש לחוש להרבה מכשולים שיבא לקדש ח"ו איסור ערוה או שניה כאשר קרה בזמנינו
It was not for no reason that the generations have not assumed the role of marriage conductor without the allowance of the Rabbi, because there are many issues that can arise of forbidden unions (translation mine)
that this is specifically for a relative who knows the family and knows if there are any forbidden or prior relationships, and if anything "strange" happens a Rabbi could be called.
As an aside, the "orchestrator" does not have to have the title "Rabbi" and does not have to have a piece of paper saying he received ordination. He just has to know all of the laws, which is possible without being titled "Rabbi." However, as a general rule, today those who would qualify as having the prerequisite knowledge are Rabbis.
In a nutshell: in U.S. state laws, a couple is "married" by a clergyman. (Alternatively, they could go to a justice of the peace or the like.) The laws vary from state to state as to what's called a clergyman; for instance, in New York City the congregation would sign a letter stating that he is "a pastor or associate pastor."
From the perspective of Jewish law (Orthodox), the marriage happens when he gives her something valuable that he owns (usually a ring) in front of two "kosher" witnesses. The witnesses need to be Orthodox-observant men, so often at a wedding with few Orthodox attendees, the rabbi is also one of the witnesses.
About a thousand years ago people were making all sorts of mistakes regarding weddings (e.g. using a borrowed ring, or related witnesses), so a policy was enacted to have a local rabbi present -- just to make sure everything is being done by the book. Thus the practice today to have an Orthodox rabbi "officiating" -- just making sure everything is done correctly. If a couple got it all right and no formal rabbi was there, the wedding would still be valid. (Though from a US state perspective, they'd have to go to a judge -- or a formal rabbi -- to sign their papers afterwards.)
I've attended many Orthodox weddings where, though rabbis may or may not have been present, they did not conduct the service.
I've personally never heard of a wedding requiring anything but any one of the following with two kosher witnesses present:
- exchange of an article of estimable monetary worth
- exchange of a contract
- some time alone for the bride and groom
In a Jewish context, the officiant doesn't "marry" the bride and groom - the groom "buys" the bride and the bride accepts. Two witnesses also sign a document stating that they saw it happen. Then the bride and groom go off to the yichud room to spend some time alone -- their aloneness witnessed, usually, by two different witnesses.