I actually receive customer feedback that the site is very slow to load and at times it is extremely slow, so slow I can't even access our own backoffic as it times out.
It does seem to be ok now though.
Then that is a far better reason to look into speeding up your site.
Cheap over-subscribed shared hosting could be the problem, and intermittent times when it slows down could be an indicator of other sites on the server hogging the resources, which might be the case with cheap shared hosting or low quality shared hosting which isn't performance monitored correctly by the server owner.
Other simple things like backups, depending on what backup software is used, might cause the system to slow down at certain times, so it might be worth finding out about this. For example if you have scheduled 4 backups a day and some of those backups take place during working hours and those backups affect performance, then this is an issue which could be easily addressed by scheduling backups out of hours. Some backup software can cause database locks.
Nope, not magento. It is their own ecommerce solution in asp.net
The problem could also be with server code/database design rather than a simple shared hosting issue, e.g. database locks slowing the system down when multiple users are accessing the site, compounded perhaps by bad coding (issuing too many SQL queries on a row by row basis, when one query would suffice, wrong database cursor types, using more connections than necessary, etc). So it is not always a simple case of fixing the problem by changing hosting or increasing the hosting spec. This scenario is less likely with tried and tested off-the-shelf ecommerce systems where scalability limits are known, but if as you say the system is fully bespoke, then it is an unknown in the equation which people here cannot really comment upon for sure, since we do not know the system.
I have tried to address this to our web developers who host the site who simply say an upgrade to an non-shared website is appropriate in that case.
There is not that much traffic
Personally, I think it is overkill to say that an ecommerce site that is not getting much traffic requires a non-shared/dedicated host. Certainly do not place an ecommerce site on the cheapest of shared hosting, sharing with thousands of other sites, but better quality shared hosting should be OK for a low traffic ecommerce site. If it isn't OK, then this suggests that something is wrong with the underlying ecommerce system or configuration.
Have your developers said why they think it is slow? and do you know what other websites are running on the same server? (are they all the developer's sites, or are there other sites too.) Do you know if the database behind the site is MS SQL Server or MySQL or MS Access?
p.s. I also just noticed a delay on your site, when the site was 'thinking' before popping up a survey screen from the home page. The site does seem slower now (4:55pm) than earlier when I first visited it.