These two words appeared frequently in a passage, for example: "There is no reason to limit the number of seats in a row other than by travel distance, provided there is an adequate gangway at both ends of the row so that escape is possible in two directions and the seatway is generous." To my understanding, seatway is the horizontal passage between rows of seats while gangway is the vertical passage. But I can't find much explanation about "seatway" on the Internet. Can't be sure whether I am right or not.
"Seatway" would appear to be a term of art for the distance between rows or tiers of seats. Consult the following manual (complete with diagrams) from the British Association Of Seating Equipment Suppliers. The manual defines gangway as
The access space internally on a tiered seating block that provides the means of distributing people on a seating area and for evacuation from the area.
What the OP calls the "gangway at both ends", the BAOSES calls the lateral gangway or vertical aisle.