Align all text in bullet

Your code is very complex. For the problem in question, I simply defined a custom SubHeadingListenvironment, based on itemize with the relevant parameters.

Unrelated: hyperref should be loaded as the last package, with very few exceptions (most notably cleveref). Load xcolor rather then color. The usenames option is obsolete, as it has been the default for quite a few years. It is unnecessary to load palatino for the footer since it is loaded for the whole document.

 documentclass[letterpaper,11pt]{article}
usepackage{latexsym}
usepackage[empty]{fullpage}
usepackage{titlesec}
usepackage{marvosym}
usepackage[dvipsnames]{xcolor}
usepackage{verbatim}
usepackage{enumitem}
usepackage{fancyhdr}
usepackage{graphicx}
usepackage{palatino}
usepackage{lastpage}
usepackage[pdftex]{hyperref}

pagestyle{fancy}
fancyhf{} % clear all header and footer fields
fancyfoot{}
renewcommand{headrulewidth}{0pt}
renewcommand{footrulewidth}{0pt}

cfoot{{footnotesize{Page thepage hspace{1pt} of pageref{LastPage}}}}


urlstyle{same}

raggedbottom
raggedright
setlength{tabcolsep}{0in}

% Sections formatting
titleformat{section}{
  vspace{-4pt}scshaperaggedrightlarge
}{}{0em}{}[color{blue}titlerule vspace{-5pt}]

%-------------------------
% Custom commands
newcommand{resumeItem}[2]{
  itemsmall{
    textbf{#1}{ #2 vspace{-2pt}}
  }
}

newcommand{resumeSubheading}[4]{
    setlength{itemindent}{-1em}vspace{-1pt}item[]
    begin{tabular*}{0.97textwidth}{l@{extracolsep{fill}}r}
      textbf{#1} & #2 
      textit{small#3} & textit{small #4} 
    end{tabular*}vspace{-5pt}
}

newcommand{resumeSubItem}[2]{resumeItem{#1}{#2}vspace{-4pt}}

renewcommand{labelitemii}{$circ$}

newcommand{resumeSubHeadingListStart}{begin{itemize}[leftmargin=*]}
newcommand{resumeSubHeadingListEnd}{end{itemize}}
newcommand{resumeItemListStart}{begin{itemize}}
newcommand{resumeItemListEnd}{end{itemize}vspace{-5pt}}

%-------------------------------------------

newlist{SubHeadingList}{itemize}{1}
setlist[SubHeadingList]{label =textbullet, wide=0pt, leftmargin=*, after=vspace{-5pt}}
%%%%%% CV STARTS HERE %%%%%%%%%%%%%%%%%%%%%%%%%%%%


begin{document}

%----------HEADING-----------------
begin{tabular*}{textwidth}{l@{extracolsep{fill}}r}
  {fontfamily{qag}selectfont{textcolor{blue}{Large textbf{Mousumi Das}}}} & includegraphics[width=4cm]{pepe-le-pew2} 
end{tabular*}
Morrisville, NC


%--------PROGRAMMING SKILLS------------
section {textcolor{blue}{fontfamily{qag}selectfontLarge{textbf{Qualification Summary}}}}
 begin{SubHeadingList}
item Statistical Programmer with 6+ years of experience in CRO environments supporting biopharmaceutical clients.

 item
     Extensive experience producing safety and efficacy Tables, Figures, Listings (TFL) as per statistical analysis plan (SAP) and mock shells.

 end{SubHeadingList}
vspace{-8px}

end{document}