Is the prayer of one who performs tayammum instead of ablution permissible due to obsessive-compulsive disorder (waswas), chronic sensitivity that causes wounds, and the occasional unavailability of water? And is she required to make up missed prayers, knowing that she is unable to do so due to headaches and dizziness?

The cure for waswas (whispers/obsessive thoughts) is to ignore them and pay them no mind. Tayammum (dry ablution) is only permissible when water is unavailable or its use would cause harm. If using water would harm you, then wash what does not harm you, wipe the rest, and if wiping would harm you, then perform tayammum for it. So, whoever is harmed by using water on some limbs but not others should wash what is possible and perform tayammum for the rest. As for making up missed prayers, it is permissible to adopt the lenient opinion. Regarding physical pains, you can consult the 'Consultations' section of the website or trustworthy doctors.