As sweden is 'cashless', I am wondering if I can avoid currency exchanges altogether by purchasing (and topping up) the SL access card with my credit card. Is it possible? Or should I fetch some cash to get the cards?
All places that sell SL Access Cards accept credit/debit cards. Almost all accept cash.
You can't buy tickets on busses or on trains. SL has a help page about buying tickets.
Ways of getting phyical tickets
You will find self-service machines at all larger stations and select smaller ones, they only accept credit/debit cards.
All the entrances to train & metro stations are manned and you can buy tickets and access cards there.
Many kiosks, or "tobacco shops" as they usually call themselves, sell SL Tickets and Access Cards without markup. Look for an SL Flag or some other mentioning of "SL Tickets".
There are also a few SL Center's in Stockholm. There are 2 in Stockholm Central but there are also many Pressbyrån there that also sell the tickets. At SL Center they can answer any question that you might have about the public transportation system as well as sell you tickets.
Where to find these places
Go to SL.se/en and use the map. Choose Sales points and find nearby place.
Yes, I bought an SL Access card+travelcard from a manned metro gate with a credit card in February 2019. In May 2018 I bought one with value from the SL Center with a credit card.
There is also plenty of retailers that sells SL Access cards and offer topups, probably all of them accept credit cards, at least Pressbyrån does. Check the map here: https://sl.se/en/ Use the icon on the top right and tick sales points instead of stations. Green icons shows where you can buy and top up SL Access.
Visa and Mastercard are the universally accepted cards in Sweden.