Well, there's an incredibly specific question.
I do not live in Eskilstuna, but I think I know a bit about how things work and did a bit of googling in Swedish. Generally, these are small shops that often don't have a website, let alone a webshop. It's also much more common to go pick up cakes yourself instead of doing deliveries. You'd have to contact some "konditorier" on Facebook or something, here's a list of konditorier in Eskilstuna.
The closest thing to what you want that I found would be Konditori Janeling, which has a webshop and apparently delivers stuff. But their payment system is Klarna, which I believe is Sweden only. Maybe you could shoot them a mail to see if they could offer an alternative.
Maybe finding a proxy for delivery would work if nothing else. Here's a Facebook group for people in Eskilstuna posting random stuff, you could also go to the Sweden subreddit and ask there, people on there can be pretty helpful.
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Hi, Anyone here from Sweden can tell me is there any good online shop that will deliver a cake to a Birthday Person living in Sweden, Eskilstuna?
And accept PayPal. Thanks.
Edit:
Placed an order on a website called dinTarta.
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