Each antenna is able to excite almost any mode, and will. The only modes that won't be excited by an antenna are those for which the antenna is at a node of the mode, a point where the standing wave of the mode has no amplitude.
Each antenna is at a different point in the cavity, so will excite different modes in different ratios.
If there only existed one mode how will the em field look inside the container for each of the 4 antennas?
The only way for there to be only one mode is if the cavity is much smaller, comparable to the half wavelength.
I'm am basically working backward in that I already have a cavity resonator and the frequency.
No, you don't have a cavity resonator, at least not what engineers would call a cavity resonator. You have a honking great box. A cavity resonator supports only one mode, it's built small enough or operated at the right frequency, to do that. You have a large box supporting many, many modes, most of which will be excited at essentially unpredictable levels by any radiation coming in from the antennae.
There is no need for you to visualise the modes in your box, all you need to know is that there are a lot of them. There are so many that the pattern you actually get will be unpredictable, given the accuracy with which you can characterise the box and the antennae.
A standard domestic microwave oven cavity is multimode. It will contain a number of high field and low field regions, which will result in over-cooked and cold food respectively. The three solutions to this are a) a turntable, to try to move the food through a variety of regions, b) a mode stirrer (very few have these) which is a metal vane rotating in the cavity to change the electrical geometry (roughly equivalent to your multiple antennae) and c) a single mode cavity with predictable power levels, these are only used in industrial applications where the cost of low frequency RF power generation is acceptable.