How do I fix this error with quotation marks?

Your string does not make sense - I am providing the options based on what you provided but I cannot tell what you are trying to do: there is a lone ' in the middle of it - what is the output you are aiming for?
Option 1
Escape with \

LEFT JOIN ' . $supplier['table'] . 'Cats\'\' USING (Category)');

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Option 2
Use double quotes for the string

somefunction ("SELECT ...LEFT JOIN " . $supplier['table'] . "Cats' USING (Category)");

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Option 2a
Use double quotes for the string and embed var with { }

somefunction ("SELECT ...LEFT JOIN {$supplier['table']} Cats' USING (Category)");

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Option 3 [PREFERRED]
Use HEREDOC. Allows use of both single and double quotes without escaping and embedding of variables

$query = <<< QUERY
SELECT ...
LEFT JOIN {$supplier['table']} Cats' USING (Category)
QUERY;

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Julian,

I am unable to get the code to work with the above examples.  Here is the full string for the Left Join.  

$query3 = $conn2->query("SELECT DISTINCT SubCategory, Category, SubCatID FROM " . $supplier['table'] . ".SubCats LEFT JOIN " . $supplier['table'] . ".Cats USING (Category)");

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Thanks!

You gave us this (double single quotes - was that supposed to be single double quotes?)

LEFT JOIN ' . $supplier['table'] . 'Cats'' USING (Category)');

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Now you give this

$query3 = $conn2->query("SELECT DISTINCT SubCategory, Category, SubCatID FROM " . $supplier['table'] . ".SubCats LEFT JOIN " . $supplier['table'] . ".Cats USING (Category)");

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That's not the same code
If you used the suggestion above and put {} around your strings and not use concatenation you would have got this

$query3 = $conn2->query("SELECT DISTINCT SubCategory, Category, SubCatID FROM {$supplier['table']}.SubCats LEFT JOIN {$supplier['table']}.Cats USING (Category)");

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My preference would have been using HEREDOC as follows

$sql = <<< QUERY
SELECT DISTINCT
  SubCategory, 
  Category, 
  SubCatID 
FROM 
   `{$supplier['table']}.SubCats` LEFT JOIN `{$supplier['table']}.Cats`
USING
  (Category)
QUERY;
$query3 = $conn2->query($sql);

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Much easier to see what is going on.

BTW $supplier['table'] is misleading - is that meant to be a database reference with .cats and .subcats being the actual tables?