Right now there are images of the US Senate being occupied by protestors, which obviously makes it impossible for the Senators to gather there. Is there an official designated spot for situations like this where Senators could gather in safety and conduct their business? Or perhaps even multiple alternative secure locations?
According to an NPR article, a secret bunker (Project Greek Island) was built in West Virginia's Greenbrier Resort in the late 1950s. It was large enough for both chambers of Congress to hold sessions, contained 1100 beds and a 6 month supply of food, and was to be used by Congress in the event of a nuclear war or some other emergency. It was exposed by the Washington Post in 1992, at which point it was decommissioned.
The article says the bunker would likely still be used today as the secret location for Congress if it hadn't been exposed, and that Congress's current secret location is, well, obviously a secret:
Today, part of the bunker is a tourist attraction. Another part is used as a secure data storage facility. Had it not been exposed in 1992, there's a good chance this would still be the secret home of the U.S. Congress.
But now that secret home is somewhere else. And, like the last one, just a handful of people know where it is. Post reporter Bill Arkin is one of them, and he's not saying.
Mount Weather Emergency Operations Center in Virginia can accept leading congressional figures - and it did, by helicopter, after the attacks of 11 September 2001 - but as far as I know it is not equipped as a location at which the whole business of Congress, including chamber debates, can be conducted.
The short answer: there are such facilities, but where they are and what capabilities they afford is unclear.
The long answer: the US Congress is not bound to any particular location, and could use any of a number of facilities that are part of the US government continuity of operations apparatus, but which exactly is not public knowledge. According to Garrett M. Graff, "the facilities in use today exist in a strange netherworld where they are highly classified but also mostly mundane government offices, albeit underground." Some, like Fort McNair (to which congressional leadership was taken yesterday), might only serve as temporary secure accommodations for a select few; others, like Mount Weather (mentioned by @ruffle, where congressional leadership was evacuated during 9/11), might serve as long term accommodations to guarantee the presidential line of succession, and the basic functioning of Congress during wartime. Others yet, like the Greenbrier (mentioned by @pacoverflow), might be equipped to handle a fully-attended Congress over a prolonged period of time.
Starting in 2020, as a result of the COVID-19 pandemic, Congress has another option: convening remotely from multiple locations by video link. The Continuity of Congress website (from NGO Demand Progress) documents the progress made in that direction.
Mind that having a place to convene in is only a part of the equation: the evacuation, activation and operation procedures must be fast and reliable enough that Congress can continue its business more or less smoothly. If Congress is working under a legal deadline, or if a majority of members did not make it to the evacuation point (a scenario considered by the Continuity of Government Commission in 2003), the US might still find itself facing a national emergency and a constitutional crisis.