Is it possible to add fluoride to my tap water?
I know this has gotten a bit off topic, but in an attempt to answer the question - according to my wife, a hygienist - your best bet is to use:

Fluoridated toothpaste
Take a fluoride supplement


However, I am specifically asking about ways to add fluoride to tap water and I am not asking about these alternative methods.

Too much fluoride can harm you. After researching this morning, I don't see a way to add it to actual tap water coming out of a faucet as your looking for. Trying to calculate water pressure, flow rate, storage and release time of fluoride tablets would be rather difficult I think. Similar to trying to add a multi vitamin to tap water stream. To little and it'll be ineffective; to much and it would make you sick.
This article may help

According to the CDC, &quot;if fluoride levels in your drinking water are lower than 0.7 mg/L, your child's dentist or pediatrician should evaluate whether your child could benefit from daily fluoride supplements.&quot;  The American Dental Association has a standard dosing schedule for fluoride supplements. Fluoride can be given in liquid form or in tablets and is usually taken twice a day and can be mixed with juice, milk, water, or food.
In your conversation with your dentist, you may find that supplements are not necessary if your child drinks a lot of tap water at school or a day care, and if your toothpaste has fluoride and is used on a regular basis. Have an open conversation with your dentist about how to get enough fluoride if you are drinking mostly bottled water, or if you use any alternative to tap water.

Ultimately, discuss this with your hygienist (it's more of the hygienist's responsibilty to answer this then the dentist's) and see if they can recommend a treatment plan or alternatives outside of the dental office.