Antonin, to understand if the harvester is working right, you can look at the Ship List tab, in the Naval Organization window.
There, you can see the fuel level stored in the ship(s).
If this value increases after some weeks the harvester is near the gas giant with Sorium, then the harvester is producing fuel and consuming Sorium.
You can also look at the level of Sorium of the planet: if it goes down, it is being used to produce fuel.
A couple of observations about the design of your ship.
1.
To transfer its fuel to a ship, the harvester must have a Refuelling System. You can research larger and large ones as Logistics topics.
2.
As it is a military ship, it has shorter Maint Life and Deployment Time than a commercial one.
Even more, in your design, these parameters are very short.
But, to fill the fuel tanks, this ship needs a bit more than 9 months (= 12*500k/640k).
Instead, those short parameters generate frequent maintenance problems, well in advance of 3 months, and the ship will risk to explode, if not serviced, and the tanks are far from being full.
So, you should keep the harvester a commercial vessel, i.e.: don't use a military engine, don't install neither active sensors nor large passive ones, and install an engineering space.
An example of my harvesting station:
FHS-01 Vasachl Harv St 001 (Vasachl Harv St class Fuel Harvester Station) 30,425 tons 130 Crew 543 BP TCS 608 TH 0 EM 0
1 km/s Armour 1-86 Shields 0-0 HTK 64 Sensors 0/0/0/0 DCR 1-0 PPV 0
MSP 161 Max Repair 30 MSP
Centurion Control Rating 1 BRG
Intended Deployment Time: 24 months
Fuel Harvester: 10 modules producing 1,200,000 litres per annum
Fuel Capacity 4,000,000 Litres Range N/A
Refuelling Capability: 50,000 litres per hour Complete Refuel 80 hours
This design is classed as a Commercial Vessel for maintenance purposes
This design is classed as a Fuel Harvester for auto-assignment purposesAnd an example of harvesting ship, jump capable too, so it can follow fleets around, if needed, and with small sensors, just to avoid surprises

:
FH-12 Gloo Harv 007 (Gloo Harv class Fuel Harvester) 39,161 tons 210 Crew 1,372.1 BP TCS 783 TH 134 EM 0
1430 km/s JR 3-50(C) Armour 1-102 Shields 0-0 HTK 87 Sensors 18/18/0/0 DCR 1-0 PPV 0
MSP 21 Max Repair 280 MSP
Trierarch Control Rating 1 BRG
Intended Deployment Time: 24 months
Fuel Harvester: 10 modules producing 1,200,000 litres per annum
JC40K Commercial Jump Drive Max Ship Size 40000 tons Distance 50k km Squadron Size 3
Commercial Inertial Fusion Drive EP560.00 (2) Power 1120 Fuel Use 0.86% Signature 67.20 Explosion 4%
Fuel Capacity 3,000,000 Litres Range 1,612.3 billion km (13049 days at full power)
Refuelling Capability: 200,000 litres per hour Complete Refuel 15 hours
EM Sensor EM1.0-18.0 (20%) (1) Sensitivity 18 Detect Sig Strength 1000: 33.5m km
Thermal Sensor TH1.0-18.0 (20%) (1) Sensitivity 18 Detect Sig Strength 1000: 33.5m km
This design is classed as a Commercial Vessel for maintenance purposes
This design is classed as a Fuel Harvester for auto-assignment purposesI am now prefering ships, because this frees my tankers to support fleets and planets around.
But this is just my personal taste, for the match I am playing now.