Specifically, I am tying to convert the names of Thor's goats Tanngrisnir and Tanngnjóstr into runic text. I have found several charts/translators online to convert English characters to Elder Futhark, but is it this simple or is there some nuance to it?
There is a lot of nuance to it. I have some experience with runology but I would never presume to try this and call it accurate. You'd need a professional runologist for that and there are probably less than 100 of those in the world.
You see, the problem is that you are not trying to transliterate letters here^* but sounds. And those sounds changed in different periods of use of the elder futhark and in different regions. There are also sounds that don't have a direct correlation in the futhark in which case a decision has to be made which rune to use or whether to omit the phoneme completely.
In order to do this authentically you have to have a deep knowledge of the Elder Futhark corpus and the solutions chosen for these problems there and there aren't many that do.
^^* ^(By the way it isn't English characters you're trying to transliterate but Old Norse ones.)
/u/wee_little_puppetman is right. Also, they wouldn't have been written in runes; Rudy Simek contests that the names are relatively recent and may well have been created by Snorri while compiling the Edda, as there are no extant runestones showing those names.
Additionally, you need to be super careful when trying to create accurate runes based on mythologies in the Edda, as a lot of it is very clearly Christian European in derivation, such as the euhemerization of the gods into heroes from classical literature, and even the concept of Ragnarök might be heavily borrowed from Christian mythologies (the earliest attestation we have of anything like the Ragnarök story is the Skarpåker stone from Sweden, dating from the early 11th century; everything else is well after that, and clearly has a Christian bent in the surrounding literature).
That said, if you really want to use elder futhark runes (which, honestly, wouldn't make sense unless you're doing pre-8th century stuff; if you're doing post-8th century stuff, younger futhark's the way to go), the names would simply be ᛏᚨᚾᚷᚱᛁᛊᚾᛁᛉ and ᛏᚨᚾᚷᚾᛃᛟᛊᛏᛉ - *Tangrisniz and *Tangjostz, as digraphic runes weren't a thing, and nominative masculine singular nouns had stem-final *-z, carrying on from PIE stem-final *-s, before mutating into stem-final -r.
See: Simek, Rudolf, Angela Hall (trans.) Dictionary of Northern Mythology. DS Brewer, 2007