Hi, I am a nationally certified foot reflexologist through the American Reflexology Certification Board (arcb.net). I also am a graduate of nail technology school.
I have given foot massages to women in public and I can assure you that nobody will think you are a freak if you have confidence. Quite the contrary, you will most likely draw the attention of other women who would like to have their feet massaged as well as other guys who wish they had the courage to do what you are doing.
Here are several options you have.
I would not worry about privacy so much and just find a comfortable place and go to work. All you need is a small bath towel to put on your knee to keep the foot lotion from ruining your pants and of course, some foot lotion. You can pick up a small container of foot lotion at any drug store. Also a small can of foot spray or a small box of baby wipes to clean her feet with. If you have no technique, I would suggest watching some foot massage\reflexology videos on youtube and perhaps practicing some of the moves before you go live. You really only need about six or eight techniques to do a good foot massage.
When it comes to the feet remember that more pressure is generally preferred to light pressure and slower movement is more relaxing than quick movements. Light, fast techniques tend to tickle. Also, remember to keep your nails short as digging in with your nails can be quite painful. Remember also to not apply the foot lotion directly to her feet. Put some foot lotion in your hands and then warm the lotion for a few seconds by rubbing your hands together. Cold lotion is not good because it will also gum up and create a nasty residue rather than soak into her feet.
Here is a trick to use. If you can get to a microwave oven, put your lotion in for about 10 seconds to warm the lotion up. You have to use your judgment and not leave the lotion in so long the lotion bottle explodes or the lotion becomes too hot to use.
Thumb slides, thumb presses, knuckle rotations up and down the sides of the feet, spinal twist, wringing the foot with both hands, rotating the toes, fist slides and rubbing the feet from side to side are enough techniques to get you going. You can google each of these with the word "reflexology" and see for yourself how to do them.
The key is to have confidence. You want smooth transitions between techniques. If you have to stop and pause and if you look and act nervous that will kill the foot massage. Have confidence that you are the man and I can tell you from my own experience that you will have more ladies to give foot massages to in the future once they see what you can do.
If you absolutely cannot do a foot massage in public you might try an empty classroom, your car or the cafeteria after everyone is done eating.
Best of luck to you.