I have what I think is a peperomia plant based on other searches here but my question is about if it's meant to be upright or grow more like a vine. Is the plant, as it grows, meant to "flop over" like in the photo here (House Plant With Round Waxy Leaves Identification) or stand up tall? Mine was very short until recently and has grown quite tall but I've also been propping up the stem as it seems too heavy and wants to topple over. I'm afraid of the stems breaking but the picture in the other post makes me think it will adjust to not being upright? Thoughts?
Healthy looking plant. Usually, the stems do trail over the edge of a pot - the trouble is, if you've kept yours staked upright for some time, those stems will now be quite stiff and not very flexible. You need to find another spot for it so it can sprawl and remove the supports. If the stems break, just cut back to above a healthy leaf - it will grow again. The only other alternative is to keep it staked, but that is not its natural growth habit and over time, it may get too tall to be supported easily in a pot.
Emily, your Peperomia is meant to be mostly upright. You can confirm this with a Google search for images of P. obtusifolia where mostly they are stiffly upright, but somewhat leaning when they get mature and start producing flower spikes. At maturity they are normally only about 6 inches tall so by normal standards yours would be about to flower and not get any taller/longer, instead producing more growth at the base.
Normally the stems, while not containing a lot of fibre to keep them upright, are thick and sturdy. One reference book (Hortus Third) describes the stems as "stout". This is usually enough to keep them upright, but if they get soft or top heavy then they have to lean and flop over due to gravity. Extended stems usually means not enough light to keep the plants compact (plants generally stretch when in low light) but in the case of Peperomia it should be accustomed by nature to low light. So it might be a matter of too moist at the roots or a bit too much fertilizer and TLC. Your plant really wants to be in warm, moist air but fairly dry at the root in quick-draining soil.