Whilst it isn't ideal that you've taken on a role, and are now asking for adjustments, it shouldn't something that you avoid. If somebody is perfectly fine, and becomes disabled whilst employed are they not allowed to ask for adjustments? As you've tagged that you're in the UK there are specific things that will apply to your employment.
Whether your phobia qualifies as a disability, which would cover you under the Equality Act 2010, is something that will need to be determined by a professional. If it is then you're protected from discrimination and entitled to adjustments to your work so that you are not disadvantaged. It's defined as:
The Equality Act says a disability is a physical or mental impairment which has a substantial and long-term adverse effect on your ability to carry out normal day- to-day activities.
What you can do is ask for Reasonable Adjustments, these are when the employer modifies your surroundings, or role, etc to make it easier for you to work. An employer has to try and do this for employees conditions, however what is "reasonable" is open to interpretation.
If you are diagnosed professionally the employer may wish to verify this through their own, or 3rd party, occupational health service. Occupational health will also inform the business of any recommendations they have in regards to reasonable adjustments.
As your request is to work remotely, and they have the facility, they may permit this. However it is worth noting that under the employer is responsible for protecting your health and safety - even when working from home. If you're doing it for prolonged periods of time then, to protect themselves, they'd need to ensure that you're using safe equipment, sitting correctly etc. All of the things they cover in the office - they'd have to cover in your home, and that costs money and requires periodic checks. Of course not all employers do that and put responsibility to the employee, but it is a risk if you injure yourself on works time.
IANAL, so you would need to speak with a professional to check your legal standing. However it's worth asking if they would be open to working from home, then if they say no you can find somewhere else, but if they say yes then you are onto a winner.