So, yes, TR did go on several extended trips during his presidency, including some to Yellowstone, but I don't know that I would really call it "going AWOL." For one thing, these trips were planned weeks, if not months, in advance. This letter from Roosevelt to John Burroughs describes how Roosevelt plans to take time to visit Yosemite and Yellowstone National Parks as part of an extended trip across the country, spending five days and two weeks, respectively - https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o283488. As you'll note, however, Roosevelt is writing this letter in mid-March 1903, so he has his trip planned and routed at least two weeks prior to actually going.

He also notes that this itinerary is settled "subject to unforeseen disaster in the Senate." This is an especially important point because at this time Congress was not meeting year-round. In 1903, for example, Congress was in session between March 5 and March 19, and then let out until business resumed on November 9. So during the summer months (and even late spring, when Roosevelt went on this particular trip), there wouldn't be new legislation coming for him to act on.

The Western Trip itself, I should mention, was seen more or less as part of the presidency. This article, from early April 1903, says not only that Roosevelt was using this trip to generate support for his policies from the Western states, but also to generate early support for himself as a candidate for the 1904 elections - https://www.newspapers.com/clip/87414492/ - making the trip as a whole not some frivolous journey away from the White House, but a way for him to connect with the broader United States than was usually possible from Washington, D.C. and New York.

As far as going into the parks themselves, it would appear that Roosvelt's insistence on leaving behind the press and nearly all of his aides did occasionally cause some worry, as noted by author Chris Epting in a talk he gave at the TR Inaugural Site in 2016 - https://www.trsite.org/blog/2016/08/08/tr-in-california-the-whistle-stop-tour-that-changed-america - where some inclement weather was a cause for hesitation. Nevertheless, (jumping back to Yellowstone) reading through John Burrough's account of his time in the park with TR, it sounds like although TR insisted on leaving the press behind, the sort of itinerary he followed in the park would allow people to know roughly where he was and get in contact with him if it were absolutely necessary - https://www.theatlantic.com/magazine/archive/1906/05/camping-with-president-theodore-roosevelt/307260/. Burroughs references journeying to specific, known sites to make camp (in addition to occasionally staying near houses or barracks), and mentions at one point being joined by a mounted rider who turned out to be a government scout.

Related to this, it appears that, if need be, Roosevelt had the ability to receive and send communications. This article writes that "Roosevelt will be in almost daily communication with Secretary Loeb at Cinnabar, but nothing except of the utmost importance will be referred to him" ("Secretary Loeb" being William Loeb, Roosevelt's personal secretary, not a cabinet secretary) - https://www.newspapers.com/clip/87415884/; https://www.newspapers.com/clip/87415921/ (sorry that clipping is in two parts--it's just how it worked with the page layout).

I'll see if there's any further information I can add tomorrow--I don't have access to all my TR-related books from home. In any case, I hope that gives at least some context for how Roosevelt could have been out of the office for so long.

EDIT: A lot of what I talked about above deals with Roosevelt's 1903 trip, but he also had a significant hunting trip to Colorado in 1905 that might be worth mentioning, as it's another instance of Roosevelt being out of office for an extended period of time.

Looking at this trip shows many of the same considerations as his 1903 venture: Roosevelt waited to leave until after Congress closed its session (which appears to have adjourned from March 4, 1905-December 4, 1905), and William Loeb stayed relatively nearby at a sort of "base camp" in case of urgent news (https://www.newspapers.com/clip/87429761/).

Unlike his 1903 trip, however, Roosevelt seems to have more explicitly left someone in charge in Washington while he was away during this trip. A number of political cartoons (see bottom right cartoon here - https://www.loc.gov/resource/mss38299.mss38299-477_0001_0741/?sp=139 - or left cartoon here - https://www.loc.gov/resource/mss38299.mss38299-477_0001_0741/?sp=148) reference Roosevelt as having left Secretary of War William H. Taft in charge of "keeping the lid on things" in D.C.

Edmund Morris in Theodore Rex writes that "Rather than leave the White House in charge of Vice President Fairbanks, who had been relegated to near-total obscurity since the Inauguration, Roosevelt assigned crisis management powers to William Howard Taft. 'I am not entirely satisfied with the foreign situation,' he admitted to Hay, 'but there isn't anything of sufficient importance to warrant my staying.'" ... "At any rate, Taft could be relied on. The President stayed in Washington just long enough to hand him a new, reorganized Isthmian Canal Commission. Then he quit town, leaving instructions that he be wired at any change in the international situation." (380)

Morris goes on to write how Roosevelt received and sent several telegrams from "Camp Roosevelt" in Colorado relating to Roosevelt's offer to mediate between Russia and Japan to end the Russo-Japanese War. In the same chapter of Theodore Rex, Morris recounts that in late April, while Roosevelt was out on his hunting trip away from his main camp area, "a telegram in cipher from Taft arrived by special messenger at the White House communications center in Glennwood Springs. After it was decoded overnight, William Loeb found that it contained the text of a secret cable from Baron Jutaro Komura, the Japanese Foreign Minister to Takahira" ... "Loeb felt unable to trust any messenger with such a document, and decided to deliver it himself. He took a train to New Castle, then hired a mustang and a horse wrangler and ascended the mountain there. Arriving at Roosevelt's camp late that afternoon, he handed the telegram over." (382) Rather than returning, Roosevelt dictated a telegram for Loeb to send to Taft, and kept hunting for several more days. So again, while he was certainly out of office for weeks at a time, it does not seem that he was necessarily out of contact with what was going on (at least not for extremely extended periods of time).