Imagine you take two glasses and place them upside-down on a table. One covering a certain amount of water (just poured on the table) and the other being empty. How does relative humidity change in the two glasses?
If the temperature of the room remains constant, the relative humidity of the gas mixture in the glass not covering water will remain the same for the following reason.
As you have indicated, the relative humidity (RH) is defined as
$$RH=\frac{p_v}{p_{sat}}$$
Since the amount of water molecules under the glass is unchanged, for the RH to change the saturation pressure ($p_{sat}$) needs to change. The latter depends on temperature (it decreases when temperature decrease and vice versa).
For the glass covering some water, assuming the water is at room temperature, the relative humidity will increase. This is because the water will evaporate increasing the partial pressure of the gaseous $H_{2}O$, up to a maximum of 100 %, depending on the amount of water covered and the volume of the glass.
Now, regarding the lead in questions.
What physical process determines the vapor pressure $p_{vap}$ in a given state (p,V,T) in a closed container? What balance of forces (pressures) fixes the value of $p_{vap}$?
The physical process involves evaporation. The amount of water that can evaporate is limited to the amount of water available to evaporate, the space available for it to fill (volume), and the temperature of the air in the space (which determines the saturation pressure of the air). The greater each of these is the more water in the gaseous phase that can be accommodated. As the water vaporizes the vapor pressure above the surface of the water increases slowing the rate of evaporation. Evaporation balances condensation at the surface when the relative humidity of the container reaches 100 %.
What does vapour pressure $p_{vap}$ depends on? Is it a property of water only and its state or depends on environmental conditions like, say, air pressure as well?
The partial pressure of the gaseous $H_{2}O$ is the pressure that it alone would exert if all the other gases in the container were removed. It is a property of the water (saturation temperature and pressure of water). It depends on the environment since the temperature of the environment determines the saturation pressure. In the atmosphere at sea level the partial pressure of the water vapor is very small. For example, for a temperature of 20 C and relative humidity of 50% the partial pressure of the water vapor is about only about 1% of the total atmospheric pressure.
ADDENDUM
Overall your questions concern the subject of Psychrometrics, the study of water vapor in atmospheric air. A convenient reference showing the relationship between thermodynamic variables pertaining to the partial pressure of gaseous water vapor is a Psychrometric Chart. The chart assumes one is dealing with standard atmospheric pressure. The chart includes the following properties:
• Relative humidity
• Humidity ratio
• Dry bulb temperature
• Wet bulb temperature
• Saturation temperature (dew point) (saturation pressure)
• Specific enthalpy
• Specific volume
From the chart, given values of any two of the above properties any of the remaining properties can be determined, or at least estimated. While the partial pressure of the gaseous water vapor $P_{vap}$ is not on the chart, it can be calculated from the relative humidity and dry bulb temperature properties on the chart.
Hope this helps.