I found that some websites refuse to acknowledge that plusses are valid.

I'm not saying that they won't, but they're non-compliant then.

https://en.wikipedia.org/wiki/Email_address#Local-part

The format of an email address is local-part@domain, where the local-part may be up to 64 octets long and the domain may have a maximum of 255 octets.[5] The formal definitions are in RFC 5322 (sections 3.2.3 and 3.4.1) and RFC 5321—with a more readable form given in the informational RFC 3696 (written by J. Klensin, the author of RFC 5321) and the associated errata.

Local-part

The local-part of the email address may be unquoted or may be enclosed in quotation marks.

If unquoted, it may use any of these ASCII characters:

I don't want to try to escape the following for Markdown, so I'm just gonna dump it in a blockquote:

uppercase and lowercase Latin letters A to Z and a to z
digits 0 to 9
printable characters !#$%&'*+-/=?^_`{|}~
dot ., provided that it is not the first or last character and provided also that it does not appear consecutively (e.g., John..Doe@example.com is not allowed).[8]

If quoted, it may contain Space, Horizontal Tab (HT), any ASCII graphic except Backslash and Quote and a quoted-pair consisting of a Backslash followed by HT, Space or any ASCII graphic; it may also be split between lines anywhere that HT or Space appears. In contrast to unquoted local-parts, the addresses ".John.Doe"@example.com, "John.Doe."@example.com and "John..Doe"@example.com are allowed.