@Megan Hirlehey I generally agree with @Ashish Acharya, but let me address a few point/questions that you brought up in your post:
"My question is, what are the tax/legal implications of taking withdrawal (not a loan, this money will not be returned to my own LLC) from my LLC to use as my half of the seed money for the new LLC?"
This actual depends on the tax treatment of your LLC. As @Ashish Acharya pointed out, by default, an LLC is disregarded as a separate entity form its owner, if it has only one owner, which based on your post I assume is the case. Alternatively, you can elect to treat your LLC as a corporation for tax purposes, which may have some tax implications. In general a distribution from a disregarded entity is not considered income to the person receiving the distribution because the owner is treated as owning directly all the assets of the entity, so from a tax perspective this is the same as if you withdrew money from your own bank account. Distributions from corporations however, may be taxable to the person receiving the distribution if the corporation has positive earnings for the year.
You should confirm that your LLC is disregarded for tax purposes.
"I have not generated a profit yet in my own LLC, so the money that I would be taking from it is basically a portion of the seed money for that account. Would I have to pay any kind of tax on this since it isn't "income" or if I do, will it cancel out as the money is going right into another investment?"
As indicated above, distributions are treated differently depending on how your LLC is treated for tax purposes. As you brought up "income", I think this is a good time to mention that if you generate positive taxable income as adisregarded entity you will pay tax on that in the year earned, regardless of distributions. However, if you LLC is treated as a corporation, the LLC would pay tax on the income in the year earned. If you then made a distribution out of the LLC taxable as a corporation you may have dividend income, which is then taxable to you. This is what most refer to as "double taxation". For this reason, taxable corporations are generally inefficient form a tax perspective for US residents. Foreigners may be able to benefit from using structures that income taxable corporations, but that is another conversation for a different day.
Regarding your "canceling out" comment, I would rephrase that as "can you deduct the cash investment in LLC 2 against the income (if any) on the distribution from LLC 1". You should note that LLC 2, as it has more than one owner, cannot be disregarded. Rather it can be either a partnership or a corporation for tax purposes. Either way, your initial contribution would be added to your basis in the membership interest, which is not currently deductible (read as: no you cannot "cancel out" the income from the distribution). You will be able to net this basis against any sales proceeds, if you sell your interest in the entity.
I will reserve comment on piercing the corporate veil as I am not an attorney.
Either way, you should (always!!) speak to your tax adviser on this (any) issue as every taxpayers personal situation is different and you may fall into traps that you are not aware of.
Good luck on your new investment and I hope your initial investment starts performing soon!