I'm incredibly naive with programming to the point where I don't even know the capabilities of what can be done, so sorry in advance if these questions are out there...
A lot of what we do is in excel and there are macros set up that reference other worksheets in excel to extract data. I know the same can be done in MATLAB, but can data be extracted (via a script) in MATLAB from a website? For example I need the US treasury rates bi-weekly. Can I get MATLAB to go to this site, take the numbers that I usually copy and paste from there, and then apply them to numerous formulas?
Also what you said about MATLAB sounds like it can be done in excel as well, besides the non-crashing part what does MATLAB offer over excel?
You wouldn't use MATLAB or Excel for this.
You'd want to use C#. You could also use phantomJS or any number of automation tools but I'll go over the .NET approach here.
You'll use HTMLAgilityPack from Nuget, you'll then grab the URL for the data you want, you'll then use HTMLAgilityPack (HAP) to navigate the DOM tree to extract the data you want.
From there you'll validate the data is in the correct format, and then you'll do whatever operations you need to perform against the data (maybe in C#, maybe not - depends on the operation and how complex it is).
At that point you'll do one of two things. You'll use NPOI or OpenXML to export your data to your Excel file, or, you'll use Microsoft.Interop.Excel to push the data through COM and into Excel itself. You can automate every function of Excel with the Microsoft.Interop library; although, I recommend against this for most tasks.
Now, if you don't know C#, or any other programming language, then you're likely better off with AutoIT.
https://www.autoitscript.com/site/
You can compartmentalize the entire process into a virtual appliance that can be tucked away and simply queried for reporting.
Can MATLAB or another program go into other programs and utilize them? IE I use an economic scenario generating software for my work where I have to go in and manually enter bond prices/volatility rates/ect and then run the stochastic generator. Could a script that I write in MATLAB or another software go into this software and automatically perform the tasks I usually do manually?
Not MATLAB, but you can do this in C# using 3 different methods.
AutoIT Scripts/Macros: This is simplest. You'd write AutoIT scripts that would take you input values (your web scrapped values), and plug them into your application. This is easy as fuck.
Old School: Direct WinAPI hooks. This would hook your applications interface and plug values directly into it either by accessing it's memory or hooking the keyboard. I wouldn't do this if I could avoid it.
Modern Way: .NET WPF UI Automation. This is the best approach for programmatic access via the .NET Framework (C#, etc), and is also well documented. There's no nasty crash-prone hooks, and it should work across frameworks and platforms on any machine running Windows XP SP3 or better. .NET Framework 4.5 needs to be installed on the machine though, AFAIK.
Unless MATLAB is somehow already in the process/workflow and I'm not understanding where, then you've no reason to introduce it as an additional complexity. Unless you're looking to replace the software you're already using for calculation; but it sounds more like you simply want to automate the tasks.