The GPL does not require distributors supply the source code to you with the product, as long as they also provide a "written offer, valid for at least three years" to give you "a complete machine-readable copy of the corresponding source code."
Do they have rights to [ask] for my personal info
If they need personal information such as your name and address in order to give you the source code according to the GPL, then yes.
Do they have rights to [not disclose] source code via a download link
Nothing in the GPL version 2 or 3 obligates the written offer to be a download link. In fact, if the software is licensed with GPL version 2, they are actually required to supply the source code "on a medium customarily used for software interchange." They are allowed to charge you for the price of the medium and shipping. For GPLv3, they can either supply the source on a medium (and also charge you for the price of the medium and shipping) or they can offer you the source via download free of charge.
Since you mention BusyBox, I assume GPLv2 applies, which means a download link alone is not a sufficient offer. Although the Internet is "a medium" commonly used for software interchange, the GPLv2 literally says that you must place the software "on a medium." A URL does not place software on a medium until someone initiates a download. In other words, the "on a medium" phrase in the Version 2 license text implies that a machine-readable physical medium (e.g. a CD, a DVD, a USB stick, an SD card, etc.) is to be provided and that the fee is to cover this physical distribution. However, if AMI were to offer a free download link in addition to offering it on a physical medium (thus fulfilling their GPLv2 obligations), that would be perfectly fine, and I suspect most people would choose the free download link.
I found [a link to a Web form] in the home directory(/home/tc):
I can confirm that the link is the only offer they provided. I checked everything that came with the laptop(manual, CD) and the product page.
According to the GPL, the distributor must "accompany" the software with the written offer. Providing a download link in a text file in a particular directory inside the product as you describe seems problematic. Why didn't they include the written offer in the written materials that normally accompany the product? If it were my machine I would be likely to simply create my own home directory and never bother looking at the /home/tc directory. The presence of this text file would not itself be a problem if it was in addition to a written offer along with the other written materials (e.g. manual, warranty information, etc.). By placing only a single text file with a URL inside a directory that some users are likely not to look at, and by not including it along with other written materials that are actually included with the product, it seems to me at that they are not trying very hard to make good on their obligation to provide a written offer.
Even if we accept that the text file and/or URL might qualify as a "written offer" (I would call it more like an "online offer" or "Internet offer") visiting the linked-to page reveals other compliance problems with the "Internet offer." The form as currently written requires that users accept an additional agreement in order to request the source:
Please complete the GPL Package Request Form. Once submitted, an AMI Representative will email you concerning your request.
GPL Text (sic)
... By clicking SUBMIT, you agree to the following LICENSE AGREEMENT regarding code that is not covered under GNU/GPL
Notice that the agreement is titled "GPL Text" but the actual text in that box is not the GPL at all. The actual GPL license requires that the written offer to the source code be made to "any third party" (GPLv2) or to "anyone who possesses the object code" (GPLv3). What if the third party does not agree to those additional non-GPL terms in the GPL Text box? If AMI want to impose additional license terms for other files (i.e. non GPL files) that are accompanied in the download package, they need to separate those files and make a GPL-only package available which can be obtained without agreeing to an additional software license. Also, they should avoid implying that their terms are part of the GPL or are related to it. They are not.
Country (For European Union residents, please visit AMI.de.) *
The AMI form asks European Union residents to visit a different Web site, written in a different language, which ultimately does not offer the source code to visitors. However, the GPLv2 requires that the written offer be made to "any third party." As the form is currently written, they seem to be excluding EU residents from their offer, which the GPL does not allow. Again, if this form were simply provided as a possible convenience (at least possibly convenient for non EU residents) in addition to an actual written offer, then there would be no problem with GPL compliance.