To make use of biblatex's localisation capabilities we can define a new bibliography string or.
\NewBibliographyString{or}
\DefineBibliographyStrings{english}{%
or = {or},
}
\DefineBibliographyStrings{german}{%
or = {oder},
}
We can then define a new command that will change the "and" in \textcite to an "or".
\newcommand{\useor}{%
\renewcommand*{\textcitedelim}{%
\iffinalcitedelim
{\ifnumgreater{\value{textcitetotal}}{2}
{\iftextcitepunct{\finalandsemicolon}{\finalandcomma}}{}%
\addspace\bibstring{or}}% here was \bibstring{and} before
{\iftextcitepunct{\addsemicolon}{\addcomma}}%
\space}%
}
The \textcite commands can then very easily be patched to use "or" instead of "and" by adding \useor to the wrapper command.
For numeric and friends that would be
\makeatletter
\DeclareCiteCommand{\textciteor}[\useor\cbx@textcite@init\cbx@textcite]
{\gdef\cbx@savedkeys{}%
\citetrackerfalse%
\pagetrackerfalse%
\DeferNextCitekeyHook%
\usebibmacro{textcite:init}}
{\ifthenelse{\iffirstcitekey\AND\value{multicitetotal}>0}
{\protected@xappto\cbx@savedcites{()(\thefield{multipostnote})}%
\global\clearfield{multipostnote}}
{}%
\xappto\cbx@savedkeys{\thefield{entrykey},}%
\iffieldequals{namehash}{\cbx@lasthash}
{}
{\stepcounter{textcitetotal}%
\savefield{namehash}{\cbx@lasthash}}}
{}
{\protected@xappto\cbx@savedcites{%
[\thefield{prenote}][\thefield{postnote}]{\cbx@savedkeys}}}
\makeatother
For authortitle/authoryear you would use
\DeclareCiteCommand{\textciteor}[\useor]
{\boolfalse{cbx:parens}}
{\usebibmacro{citeindex}%
\iffirstcitekey
{\setcounter{textcitetotal}{1}}
{\stepcounter{textcitetotal}%
\textcitedelim}%
\usebibmacro{textcite}}
{\ifbool{cbx:parens}
{\bibcloseparen\global\boolfalse{cbx:parens}}
{}}
{\usebibmacro{textcite:postnote}}
Look up the \DeclareCiteCommand{\textcite} in your .cbx file, copy the definition, change the name to \textciteor and add \useor to the wrapper command (that is the portion in square brackets just after \DeclareCiteCommand{\textciteor}, create this optional argument if its not there already).
MWE
\documentclass{article}
\usepackage{biblatex}
\addbibresource{biblatex-examples.bib}
\NewBibliographyString{or}
\DefineBibliographyStrings{english}{%
or = {or},
}
\DefineBibliographyStrings{german}{%
or = {oder},
}
\newcommand{\useor}{%
\renewcommand*{\textcitedelim}{%
\iffinalcitedelim
{\ifnumgreater{\value{textcitetotal}}{2}
{\iftextcitepunct{\finalandsemicolon}{\finalandcomma}}{}%
\addspace\bibstring{or}}
{\iftextcitepunct{\addsemicolon}{\addcomma}}%
\space}%
}
\makeatletter
\DeclareCiteCommand{\textciteor}[\useor\cbx@textcite@init\cbx@textcite]
{\gdef\cbx@savedkeys{}%
\citetrackerfalse%
\pagetrackerfalse%
\DeferNextCitekeyHook%
\usebibmacro{textcite:init}}
{\ifthenelse{\iffirstcitekey\AND\value{multicitetotal}>0}
{\protected@xappto\cbx@savedcites{()(\thefield{multipostnote})}%
\global\clearfield{multipostnote}}
{}%
\xappto\cbx@savedkeys{\thefield{entrykey},}%
\iffieldequals{namehash}{\cbx@lasthash}
{}
{\stepcounter{textcitetotal}%
\savefield{namehash}{\cbx@lasthash}}}
{}
{\protected@xappto\cbx@savedcites{%
[\thefield{prenote}][\thefield{postnote}]{\cbx@savedkeys}}}
\makeatother
\begin{document}
\ldots as presented by \textcite{shore,yoon,cicero,baez/article}.
\ldots as presented by \textciteor{shore,yoon,cicero,baez/article}.
Just to test \textcite{shore,yoon,cicero,baez/article}.
\end{document}

Of course you can also hook \useor into \AtNextCite:
Just to test \textcite{shore,yoon,cicero,baez/article}.
Just to test \AtNextCite{\useor}\textcite{shore,yoon,cicero,baez/article}.
Just to test \textcite{shore,yoon,cicero,baez/article}.
And then define
\newrobustcmd*{\textciteor}{\AtNextCite{\useor}\textcite}
thanks to @Audrey for that solution.
Regarding your BTW: biblatex only uses the Oxford/serial comma for list with more than two items.
The standard definitions are:
\newcommand*{\finalnamedelim}{%
\ifnumgreater{\value{liststop}}{2}{\finalandcomma}{}%
\addspace\bibstring{and}\space}
\newcommand*{\finallistdelim}{%
\ifnumgreater{\value{liststop}}{2}{\finalandcomma}{}%
\addspace\bibstring{and}\space}
\newcommand*{\textcitedelim}{%
\iffinalcitedelim
{\ifnumgreater{\value{textcitetotal}}{2}
{\iftextcitepunct{\finalandsemicolon}{\finalandcomma}}{}%
\addspace\bibstring{and}}
{\iftextcitepunct{\addsemicolon}{\addcomma}}%
\space}
One can of course force the \finalandcomma to appear always by
\renewcommand*{\finalnamedelim}{%
\finalandcomma\addspace\bibstring{and}\space}
and the like.