IMO you have to be upright to throw. It's how you get to an upright throwing position that has significance. The ideal path would be like a 747 taking off as opposed to a helocopter.The take off angle would be closer to 45 degrees(747) than 90 degrees(helocopter).
The key is to get your body started with a shuffling action by your right foot just before you catch the ball.It's a quick step, catch,& throw movement.This momentum step is variable depending on the hitter(L/R) & the location of the pitch.Ideally it creates positive momentum in the direction of the throw. It's not mandatory to take a momentum step. As a matter of fact, many catchers will let the low outside pitch to a RHH turn their body and they throw without a momentum step.(They still have energy going to their target) There are very few,if any, professional catchers that do not start their bodies on balls directly over the plate or on pitches in to RHH & LHH's. Momentum & a clear throwing lane are generally prerequisites for throwing runners out at second.
My guess is that you are not gaining ground to second & generating positive momentum if you are standing straight up to throw.
As the Catching Coach alluded, stance is critical. I'm not going to cover all elements of the stance but,in order to get your body started efficiently the following are some check points: 1) Feet spread shoulders width or slightly wider with both heels on the ground and toes turned out.2)Shoulders square with the pitching rubber.3)Hips at or slightly below knee level with a slight bend at the waist and your head up. 5)Hands back with your elbows above & slightly outside your thighs.6) Weight evenly distributed & flat(Not way up on your toes) with possiblely a little more weight on you left foot.(Pitcher,hitter & location dependent)The above are not absolutes. They are very dependent on your body type, athleticism, flexibility to name a few factors.
Hope some of these thoughts help.
JW