I am currently working for a small-medium sized company (~50 employees) as the sole IT staff.

Recently, we are on track to replace one expensive yet critical legacy system in favor of an application I am currently working on during my free time at work. However, its been very busy lately with issues spawning left and right so the development of the application has been lagging a lot.

How do I ask a very thrifty management to hire more programmers to help me build the new application without sounding incompetent. 

FYI. I do support of all our applications, setup and administration of the network and servers and in-house application development.

EDIT 1: Nature of the Critical Legacy System for added information

The Legacy System is the center of our operations. Basically, all our reports, notifications and data are being handled or are passing through this system. We want to replace it for a few reasons:

1. We are bleeding money paying for support that is rarely helpful (really terrible support) plus licensing cost.
2. Ridiculous cost for adding new features (500$ to adjust the width of a column in a report)
3. System Design/Architecture problems that cause the system to be slow and unscalable (Other branches find the system unusable because of how slow it is when used outside of the Office LAN)
4. Its a program made with access '97. Need I say more? It still works but its like trying to cut something with a dull blade in terms of performance