Shouldn't be a problem. If you can do it in different browsers then you can have two different people do it.
In testing this, one MSIE one Firefox I was able to make 1 reservation for Maya Grill on 3/8 at 8am in MSIE and H&V on 3/8 at 8:05 AM on Firefox. (Both cancelled immediately after they both went through).
The second test was the same day same time, to see what happens.
Interestingly enough, picking the exact same time at the same restaurant threw me the "Your selected time is no longer available" message

(go figure eh?)
Trying with a different restaurant worked just fine. (Both reservations cancelled immediately again, Disney Dining probably hates me for this type of stuff since I do it all the time

)
So, it's not an identical match, but it suggests that you should be fine.