Solar power charge controler
Short:


I charge ~= 2 x Load_hours/Sunshine_hours x Iload  Amps
Watts at 12V ~= 36 x Load_hours/Sunshine_hours x Iload   Watts
Battery capacity >= 3 x Icharge Amp.Hours
(depends on standby days required and battery life issues)


For typical US winter months


I charge = 24 x Iload  Amps
Watts = 400 x Iload Watts




Longer:

Yes. Looking at load alone and ignoring what it is for, that's a fairly standard application.
As Olin says, that's a lot of current continuously.
You did not say what voltage - lets assume 12V.

I charge = k x Load_hours/Sunshine_hours x Iload.

k is a factor that allows for charge efficiency (battery out amp-hours per in amp-hours), battery to load conversion efficiency etc. If the punch will run directly from the battery across its range then k is the battery storage efficiency factor. If you need a battery to load voltage converter then adjust k.

Sunshine hours/day varies with locality and time of year.
Most sites have down to about 2 hours/day in winter and 5+ in summer BUT eg eg New York is more like 1hr/day in Winter and Amsterdam is about 30 minutes/day in Winter and Moscow Russia is less.  This also varies daily with weather conditions.

Good solar data for your location can be found at www.gaisma.com

Example:

Use k = 2 - should be less in practice.
Assume 1.2 Amp steady - you may wish to revise this. 
Say 2 hours sun/day. 
eg Houston in December shows 2.24 sunshine hours. Close enough to 2

http://www.gaisma.com/en/location/houston-texas.html



So 
I charge = k x Load_hours/Sunshine_hours x Iload.
= 2 x 24 / 2 * 1.2 = 14.4A

If you have a 12V system then the panel max power will be at about 18V so panel Watts will be Vmp x Ip = 18 x 14.4 = 260W. A 250 Watt panel would do.

In summer this will be far more than needed. In some winter days too little.

Battery storage/day in above example is 2h x 14.4A ~= 30 Ah.
For good battery life you want substantially more capacity than daily load so the battery is not deep discharged - even with a deep discharge battery. A 100 Ah battery gives you a 100/30 ~= 3.3 capacity to demand ratio - or discharge to about 70% capacity daily. This will also provide about 3 hours operation in low sun situations.
For more safety a 200 Ah battery is better -it deep-ends on your budget and desire for system to work over long periods of low or no sun.

Depending on location etc that panel is about $1000 installed (far less DIY if your time is worth $0.
The battery is probably $200-$400 depending on brand, quality etc. Maybe more.

A lower k gives less panel size.
 If the 1.5A is only when punching then you may be able to use a MUCH MUCH lower powered system.
eg say clock took 50 mA idle and 1.2A punching for say 4 seconds. If there were 100 people punching card 2 x / day the punch related load is
people x 2 x current x second_per_punch/3600 secs/hr Ah
= 100  x 2 x 1.2 x 4 /3600 =~~~ 0.3 Ah.
Over 24 hours that's 300 mA/24 ~= 13 mA mean.
Add that to the 50 mA idle and you get about 63 mA  average
Solar charge current needed = 0.063 x 24/2 = 0.756 A
Say 1 A
20 Watt panel.
5 Ah battery.
Wow.

So knowing your location, true standby current, punching current and time that current is needed and number of punches/day is necessary for a good answer.

E &amp; OE.