The process for requesting copies of patriarchal blessings (either your own, or for a direct-line ancestor) has changed since I last wrote about it a few years ago. Here’s the current process, much easier for you at a distance because it can now be done via email.
Visit the Patriarchal Blessing Requests page at the Church History Library. This page provides general information on requesting blessings, and a link to the actual online order form. Among other details, note:
1. You can request blessings for yourself and deceased direct-line ancestors only [actually, although the website doesn't say one way or the other, I think you can also request your children's and grandchildren's blessings; I'll find out when the Library is open and update this post]. This means you cannot ask for your brothers’ and sisters’ blessings, or your aunts’ and uncles’ blessings, but you can request those for parents, grandparents, great-grandparents, and so on. (If any of these direct-line ancestors are still living, you cannot request their blessings.)
2. The linked page requests quite a bit of information. You probably won’t know it all (who knows the name of the patriarch who blessed your grandparents?), but do provide as much information as possible to assist in recognizing your blessing among the three or four million that are on file.
3. Before you can request a blessing, you need to have an LDS Account.
You already have one if you have used New Family Search recently, or bought anything from the Church’s online store, or logged onto your ward’s or stake’s website. In that case, you just need to remember your login name and your password.
If you don’t have an LDS Account, or don’t remember your password, you will need to set up an account on this page. To do that, you’ll need your membership number and the date of your confirmation (both obtainable from your ward clerk)
The Church uses the LDS Account login, in this case, to be sure you’re who you claim to be and therefore have a right to the blessings. If you have a special case or any problem — perhaps you are not LDS (and therefore have no membership number) but wish to see your LDS grandmother’s blessing — there is a telephone number on the linked page so that you can call for help.
There is no charge for blessings. They will be mailed — not emailed; please don’t ask to have them emailed. The Church asks that you requesy no more than four blessings in any month.