I have also noticed that the buttons — which I had set up to display my website’s social links — have stopped showing icons but the text is visible.
For us the social icons in your menu are displaying fine. You may want to clear your browser cache and also ensure that you’re not running ad blockers or so because some icons may get blocked by ad blockers.
I have embedded MC’s sign up form in my sidebar. Is is possible that is causing this issue?
Sure, anything is possible, but it’s not something that can be answered in general. That’s why you would need to troubleshoot and then act accordingly. If the mobile menu is showing again after you’ve removed the sign up form, then the form was the issue.
Especially if you’ve modified code, then you also may want to check if you’ve possibly introduced bugs somewhere that are now causing your issues. And if the form doesn’t work as expected, then you could try a different approach or use a different plugin for that purpose.