Let's see why it doesn't work.
The mla package says \usepackage{times} (which is not recommended), so that the default font family is, in LaTeX jargon, ptm, which corresponds to Times.
However, the Times font family that's chosen in this way has no version for the T2A encoding (that is, for Cyrillic), so when you typeset your document you get
LaTeX Font Warning: Font shape `T2A/ptm/m/n' undefined
(Font) using `T2A/cmr/m/n' instead on input line 10.
Since no font in the T2A (Cyrillic) encoding is defined for the ptm family, LaTeX switches to the default. This happens at \begin{document}. Later, when\emph` is found, the following message is issued
LaTeX Font Warning: Font shape `T2A/ptm/m/it' undefined
(Font) using `T2A/ptm/m/n' instead on input line 11.
and the reason is just the same: no T2A encoded italic font exists in the T2A family. Since T2A/ptm/m/n has already been defined as an alias for T2A/cmr/m/n, you get upright shape.
If you want to use the standard Computer Modern fonts for your document, just say, after \usepackage{mla},
\renewcommand{\sfdefault}{cmss}
\renewcommand{\rmdefault}{cmr}
\renewcommand{\ttdefault}{cmtt}
and don't use XeLaTeX. With pdflatex you must remember to add
\usepackage[utf8]{inputenc}
if you want to input cyrillic characters directly (or use another input encoding for cyrillic).
With XeLaTeX you don't load fontenc and inputenc. And you have to use fontspec to choose a system font with cyrillic glyphs coverage.
Update
There is nowadays a Times-like font that supports Cyrillic encodings, in particular T2A, also for pdflatex. The package is called tempora.
\documentclass[12pt,letterpaper]{article}
\usepackage[T2A]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{ifpdf}
\usepackage{mla}
\usepackage{tempora} % override \usepackage{times} of mla
\usepackage{setspace}
\usepackage{endnotes}
\usepackage{hyperref}
\singlespacing
\begin{document}
\emph{hi}
\end{document}
\usepackage{fontspec}works for me. – Dec 14 '12 at 02:20inputencpackage with XeTeX or LuaTeX. You should also not likely be loading thefontencpackage unless you know what you're doing. Usefontspecinstead. See Using XeLaTeX instead of pdfLaTeX. – Alan Munn Dec 14 '12 at 02:20