Adobe Form Solution
I am aware that if you subscribe to one of the Adobe products relating to PDF management that it is possible to embed a PDF form in your website which a user can fill in online. However this solution is very expensive (for charities), so I have not looked at this in any further detail.
Google Forms
Here there are several choices, but from my earlier experience there were limitations which might affect your users. Google offers a suite of programs available to people with gmail accounts. If you are a chrome book user (very simple low cost computer), you will know that while you cannot run MicroSoft Applications on your chrome book, you can run equivalent applications which are compatible with the popular Office Apps. All of this is free. But you need to be using Chrome as a browser. Not all computers will have Chrome installed. So this may block this path.
G-Suite
There is a business version as well of G Suite, which is not particularly expensive. There is the basic version of G Suite at 6$ per month, Business at $12 and Enterprise at $25. It includes secure communications, office type apps run online, calendering and various other things. Find out more here: https://gsuite.google.com/
The detail about Google Forms can be found here: https://gsuite.google.com/products/forms/
What I am not sure about is from a privacy standpoint where is the data collected and stored (you need to know this from a GDPR perspective,) whether it is encrypted, and whether copies are held somewhere. But most fundamentally, will it work in FireFox, Safari, Edge, MS Explorer etc, in other words anything apart from Chrome. If it does not, then it is a non starter.
Add a special Form Manager
There are websites that can be built exclusively for collecting data. I know there are some free ones available within the hosting, I have played with one in the past called Lime Survey, but if you are not editing things on a daily basis, then the learning curve might be a bit steep, if you need to edit something.
Use an iFrame
Some of you are now using a 3rd party system in the office to replace MESH (Home-Start users). Some of you will have your own systems. When you view a video on your website, you are looking at an iFrame or a window, through which you can see another website. In the video case it is YouTube. It is potentially possible to show the form you use to enter data into a system from a form submitted to you, but display the form directly to the user through the website. It largely depends on your form, and whether there is a way of displaying it and limiting what the end user can see of the rest of the system.
But this might be worth exploring if you are interested.
Where to from here?
The fundamentall problem is research in this area, and subsequent development is very time consuming, I am sure there are solutions out there, but so far I have not found one that fits all of the requirements. If you are interested in solving this problem let me know. If I can find enough interested parties I may look into it more detail later this year. Let mew know if you are interested.
One last point which is also potentially possible
Part of the problem for the back office is processing the incoming form data copying it across from one system to another. There are generally scripts that can be created for the receiving system which will populate a form from a document that is submitted to it. Contact Form 7 allows very precise formatting of the incoming data. It is conceivable that someone could write a script that takes the incoming form via email, identifies the fields and copies the data across automatically at the touch of a button. I cannot do all of this, but if there was an agreed standard for the submission from your website to your organisation, then someone associated with your organisation could write the backoffice processing script which would overcome the manual re-entry and save a lot of time.