I'm not sure that it is possible to travel the United States from coast to coast by one single steam powered train today, or that there are enough connected routes that a steam locomotive could run on. Assuming this isn't possible, then when was the last time period in which this could have been accomplished?
As MichaelF pointed out, it is physically possible to travel across the US on a train - there are more than enough rail lines, though today they're all for freight. As for using Amtrak lines, for the most part Amtrak does not run on rail lines that it owns. The only track that Amtrak owns outright(as far as I remember) is the Northeast Corridor, between Washington and Boston.
I'm not sure of the exact details, but most railroads have a power sharing/freight tonnage agreement with other railroads which determines how much railroads pay each other to run on their tracks.
Now, the last time you would run across the US on a steam engine entirely(we will assume here that it's not necessarily the same train) would have been probably in the early 1950's. By that point, diesels were becoming more popular. The last major railroad to convert from steam to diesel was the Norfolk and Western in the early 1960s.