I understand that there is a certain degree of taking things as a matter of faith, but there is generally a desire to "prove" the truth of one's religion. Are there any Mormon archaeologists seeking out "prove" the stories in the Book of Mormon?
TL;DR: Yes, there are some LDS archaeologists who use their Book of Mormon frame of reference to try and provide archaeological support for BOM. Much of this has been indirect support, with a general focus on Meso-American and Central American archaeology.
Disclaimer: I am not an archaeologist; therefore, I cannot provide input on the relative merits (or lack of) related to LDS archaeology. And the question I'll address is only of the efforts to "prove" truth claims through archaeology, not what claims archaeologists are substantiating or debunking. I am someone who is not LDS, but interested in LDS history.
The Book of Mormon claims that people descended from Israelites came to the Americas on ships, and the book is a record of their existence. From the earliest times of the church, there were efforts to substantiate Joseph Smith's claims about the BOM--both amateur, and professional. The 1800s were a period of growth in study around Meso-American and Central American cultures, and Mormons read eagerly about developments, hoping for evidence to support the BOM. Mormon publications such as Times and Seasons published articles giving members information on Central American archaeological expeditions performed by non-members, which were seen as giving support to the BOM's narratives (see Times and Seasons here in Times & Seasons). I have seen little evidence for LDS-sponsored archaeological trips during this time.
Brigham Young University, the LDS-affiliated university in Utah, created a permanent chair of archaeology in 1945 (see BYU Anthropology website here). Though not explicitly tasked with proving Book of Mormon history in its mission, the department and some of its associated members have worked extensively in Central America. The New World Archaeological Foundation, was founded as an officially neutral foundation but has received extensive support from Mormons and the LDS church. It was founded in 1952 and has performed research in Central America; the research is currently centered in Chiapas (see the NWAF's website here). Daniel Peterson's article in the FARMS review is an apologetic, defensive source but it does provide additional information on NWAF's history and where some of the founding funds came from (see link).
Much of the archaeological discussion and support has recently come from amateur archaeologists and a small group of scholars affiliated with LDS apologetics groups such as FAIR or FARMS (see Coe's interview, cited below). There are very few to no mentions of BOM archaeology on BYU's websites and no professors outright state their scholarly interest in "Book of Mormon archaeology." As Coe says in his Dialogue article, "...Green has termed the Church's current approach to Mormon archaeology as a "back door" one, and he is right. There is here a close parallel to the Vatican, which, while encouraging and even financing excavations in its own foundations, has carefully avoided making official statements on the remains..." And that's mostly where things stand now.
Sources:
Edit: added one link to Times and Seasons.
Yes, though not with any significant success.
11 years after the Book of Mormon was published, a non Mormon explorer named John Lloyd Stephens published his bestseller Incidents of Travel in Central America, Chiapas, and Yucatan. Stephens' book included descriptions and drawings of ancient Mayan cities and temples which were unknown before that time. This got early Mormons, including Joseph Smith himself, excited about the idea of archaeological proof of Book of Mormon accounts.
From this time forward there have been many attempts by Mormons to try to discover archaeological evidence supporting the Book of Mormon. Most of these attempts have been by amateurs and hobbyists, but some have been more serious. The most famous of these was the effort of Thomas Ferguson, who began digging in Central America in the 1950's with LDS church funding. His work ultimately proved unsuccessful and even received criticism from other LDS scholars and apologists, most notably Hugh Nibley, who argued for what has become the dominate approach of Book of Mormon apologists for decades. Rather than trying to find the Book of Mormon in the ancient world (through archaeology), Nibley argued, scholars should try to find the ancient world in the Book of Mormon (through textual and linguistic analysis).
Earlier this year the government of Oman granted permission for an archaeological team, headed by a Mormon, to dig at an isolated beach called Khor Kharfot in the southeastern Arabian Peninsula. Many Book of Mormon scholars and apologists believe that site is the most likely match for the spot where the Lehi's family set sail from at the beginning of the Book of Mormon, based on the descriptions and trajectories in the text of the Book of Mormon.
For further reading I highly suggest By the Hand of Mormon: The American Scripture That Launched a New World Religion By Terryl Givens publissed by Oxford University Press.