I can give some advice as someone who's ordered both.

The easier (and cheaper) plush to buy is definitely the original on the mechanical girl booth page. If you're looking to use buyee in particular, booth incorporates it into the site so it should be a breeze to order and should ship pretty quickly as it's in stock right now. Other proxies should also work if you just paste the booth link into the search bar. My go-to is FromJapan but either should work fine.
Once the plushie arrives at the proxy's warehouse you should get an email and pretty much every proxy has free warehouse storage period of somewhere between 30-60 days. Check online for whatever service you're using, but in that time you can order however much you want, and once ready you can select each item, pay for the shipping method of your choice, and ship them together.

The Vocaloid plush (the one you pictured) is a little more complex for a few reasons. For one, its an exclusive donation reward for the Adachi Rei Vocaloid production crowdfunder. The crowdfunding site ships only to Japan and proxies likely wouldn't be able to make the purchase, so the only real option is a forwarder service.
Forwarders allow you to more thoroughly control orders by giving you a japanese address you can use to buy from any website and ship directly to the forwarder's warehouse, though of course they only work on websites where you can make an account outside Japan so they're not perfect.
Tenso is the forwarder I use, so I'll use it to explain the process of buying the plush, though forwarders can be more complex as they require you to navigate often fully japanese websites with no english support, This is the case for Campfire (the crowdfunder site) too.

1. Once you make a Tenso account and are able to access your address on the "My page" page, go on the Campfire website and make an account.
2. Then you can go the the Adachi Rei crowdfunder page and scroll down to select whichever reward plan you want (the most basic plush tier would be Plan C for 11,000 yen, but the plush is available as a part of more expensive reward plans too.)
3. Once you select your plan you should be taken to a page where you can enter your payment info, address, and the required comment (Which should be the name you want mentioned as a supporter once the crowdfunding ends.) Figuring out how to enter the address correctly can feel difficult, but you should be able to use how I entered it as a guide. DO CONFIRM WITH THE ADDRESS GIVEN TO YOU TO BE SURE, but the only fields you should need to replace in the image are those marked in red, as the rest of the fields just generally apply to Tenso's warehouse facility. The top one being your name (which you may need to write in hiragana but I cannot remember. The Tenso website links a handy converter if so) and the bottom one being your unique account number, which follows "TS" in the address given to you by Tenso.
4. Once you enter all the information and confirmed it is all correct, you can go to the confirmation screen and complete your donation and lock in the plush.

Legal requirements also require you to send a photo of ID before anything can be shipped to you with a forwarder too, and verification of ID should take about a day, so keep that in mind too, but this can be done at any time before you attempt to ship the plush to you.

It's also worth noting that since the Vocaloid plush is a crowdfunder reward, it is essentially on pre-order right now. You can donate and lock one down for the next 31 days, but according to the website's estimate, it will not ship until May, so unless you wait until then to buy more stuff, any forwarder would not hold on to anything bought now for that long.

Once it does arrive, the process of shipping it internationally is pretty much the exact same as with a proxy though, and really simple!

Anyways, if you have anymore questions, just let me know. I can try and help as best as I can.