there are ways to work your arches and build strength in your feet. a flat foot is a foot that is too flexable, a high arched foot is one that is too rigid (not flexable enough). There are ways to work with/treat these problems, but they take time, practice, and a lot of patience.
First things first, working out your feet properly. Now, start with a good pair of wooden soled clogs. If you buy a brand like bastad, which is quality, then it will hold your foot in the right position and "work out the foot" just as if you were lifting weights or something with it. Walk for twenty minutes in these shoes each day, and then go bare foot for a while. Yes, it does tire the foot out, like any other exercise.
After you go bare foot, switch to a more comfortable, resin based clog, such as danskos. these will also hold your foot in the right position, but it's similar to being bare foot.
The next section here is about yoga and your feet. There are many yoga poses that can help you "build" your feet, and it's important to focus on the feet when you're doing them. If you go to Yoga Journal online (yogajournal.com), there is an article about feet by tias little (i believe that's his name) that gives a great discription of anatomy, how to work the feet, etc in basic standing poses. Find that alignment and work it every day. Also wander around barefoot often, and mindfully, noticing how the foot moves and what you can do to make adjustments when standing or walking.
Now, these are just to strengthen the foot and make things work. I know that this process works because in five years i've had countless yoga students with flat feet who now have beautiful arches (particularly in class and in yoga poses). Granted, they're lower than regular arches because of the bone structure, but they are there nonetheless. Similarly, my husband has had "untreatable flat feet" his whole life (his mother swore he would never have an arch) and today he has a discernable arch. He went to the foot doctor who was shocked to see the changes in his feet.
Now, when he runs or does any physical activity, he does use his orthotics. Because he is a runner, he buys special orthotic running shoes that prevent overpronation. he prefers nikes' air stasis which is available for men and women. new balance also has a shoe for this, which he says is also very good. when out and not exercising, where your comfy clogs as often as possible, and wooden clogs if you'd like. Then, when you have your discernable arch and you have a handle for how to walk, move into other shoes that you find cute or comfortable (including heels).
A word to the wise, to buy high quality shoes that are going to help you with this, you're going to lay out some cash. Consider it preventative medicine. flat feet can lead to a number of ankle, knee, hip, and spinal problems because of the way that one walks (which is improperly). So, put the money into prevention. Similarly, i do not think that bastad or dansko offer leather-free versions of their shoes. To date, i have been unable to find any vegan clogs that have the quality in the soles that these two brands have. Again, i would consider this a health-related consession. since there are no other viable alternatives, it is acceptable to buy these specific shoes for your own health and well being. also, the shoes last pretty much forever and can be refibrished for a very low cost. this is a great way to have quality shoes for a long time. (i have some pairs that are over 7 years old, two years old before i even went veg).
I hope this helps. it does take a lot of time and practice.
Be well and happy!