What is the likelihood of your average computer user figuring out how to install and put into use, Server software (not sure yet if it is a new install or an upgrade from Windows Small Biz Server 2003 which is currently installed but aged) on an existing Raid5 (I think) system. (I need to look harder but so far have not found someone within 100mi who does this professionally).
Anything can be accomplished with enough time through both personal research and asking questions of trusted folks far away.
If there's an existing server and this is going to be a new server in the environment, that helps. Because you can migrate services the old machine does one at a time, instead of feeling like you have to get it all done in a single night after-hours, or a weekend, depending on business hours.
As someone else mentioned, Small Business Server is gone and if you were using the server for certain functions, those are now (much more expensive) separate licenses.
The first step is auditing what the existing server is doing and making a list. That might be simple, or it might be a really big list. Somewhat hard to describe how to do this, depending on how many layers of that onion there are.
You'll be trying to answer the question, what specific services are running on this server?
Some examples on a typical Windows server:
Active Directory (user authentication - often with one or more Windows Domains, not to be confused with DNS domains, which can be related, but not the same thing)
DNS
DHCP
Time Services
File Shares
Print Server
Other common seen optional things in small business environments:
Terminal Services / RDP
Exchange Server (E-mail)
IIS (often serving both as a web server and/or an FTP server)
Less commonly seen but sometimes:
Windows SQL Server
And as you've mentioned:
3rd party applications
(Like your multiuser QuickBooks)
And of course:
Central control software for anti-virus / malware detection and cleaning, as well as protection of the server itself...
Backup configuration (if using Windows Backup) or Backup Application and often acting as the storage/server for same, if using 3rd party software.
Etc.
If there's a way to reach this machine or these machines from off-site, a remote Windows pro can pretty easily assist with a good audit and tell you what's going to be easy and what will be difficult.
A full audit is really badly needed if you're going to attempt a complete machine rebuild or swap in a single night. If you have the luxury of moving services piecemeal, it's a little easier and can go slower.
There will be about 5 users, (new desktops, laptops); maybe 5 programs including QB multi-user (Premier Ed.?), internet access & wifi, use of Remote Desktop, an employee time-card device and software, a bunch of printers.
New laptops/desktops - those would take a bit of time to properly configure also. I don't think it's in the scope of the server part, so I'll just stop there.
QB multi-user: That software usually just requires one of the users to also run the "server" piece. It doesn't necessarily have to be on an actual "server", but it's probably a little easier. I can't remember if getting a copy to run as a service on a real server so others can connect to it, is a pain or not. If you did put it on a user's machine, think about where the backups will go, and automating them. Losing the company's fiscal data is usually very annoying and detrimental to the business.
Internet access: Are you saying the company internet access is by way of sharing an Internet connection through this one server? Usually that job is the job of the router. Same deal with WiFi. Where the server may be involved is as a DHCP server.
As someone mentioned, using the Primary Domain Controller handle Terminal Services / Remote Desktop, for end users, isn't a great idea.
Time card device and software: Make sure it's compatible with whatever newer version of Server before committing to moving it to the new machine.
Printers: Should work fine. You'll want to either touch all the existing PCs and configure them to connect to the new printer queues or write a login script or push them out via Group Policy if you like.
You'll also want to gather up the local admin account login and password info, and the Domain Administrator password info as the initial step of the audit. You'll be needing those.
Looking at the Services control panel/administrative tools item and getting a list of running services is a good place to start first on the old machine.
Definitely consider Office365 if the Internet connection is decent and you have Exchange running today.
Not sure what to say next. Need more info.