I've been searching around and all I can find is for business owners to deduct expenses from buying equipment for their employees to use, or self-employed persons to deduct their own expenses. If my company did BYOD, can I deduct at least part of the purchase price of my laptop?
No. Unreimbursed business expenses are no longer deductible by employees as of tax year 2018:
"The Tax Cut and Jobs Act made many sweeping changes to the U.S. tax system. One of the biggest changes under this new law was the elimination of the deduction unreimbursed employee business expenses beginning with 2018 tax returns. This effectively means that employees will no longer be able to offset their taxable income by common business expenses they may incur."