There is no reason why you can't claim against them, but the procedure depends on the cover that you have under your own policy. If you have Comprehensive cover it is usual to claim under your own policy first so that the damage can be repaired without delay. Your insurance company will then write to your neighbours insurer holding them responsible and should be able to recover their outlay. Once this has been done your no claims discount will be reinstated (unless you have protected no claims discount, in which case it won't be affected). If your car was parked up when hit then liability should not be in dispute and it should be straight forward.
If you have TPFT or TP only cover then your own insurance company will not get involved and it will be up to you to claim directly from your neighbours insurers.
It might be that your neighbour wishes to pay for the damage him or herself and not involve his insurer, which is fine if you are in agreement to this.
If you are claiming via your insurer then any policy excess will be classed as an Uninsured Loss and will usually need to be recovered separately from your neighbours insurers, although nowadays some insurers will recover your excess for you at the same time they recover their outlay if you ask them. If not and you have taken out optional Legal Expenses cover this will recover your excess for you, otherwise you will have to write to your neighbours insurers yourself.
Hope this helps.