The solution you are looking for is called "artificial recharge of aquifers" or "managed underground storage of recoverable water", among several other names.
Several different techniques have been developed for this in the past, and which one is applicable depends strongly on the specifics of your local situation. I'll describe the most frequently used techniques shortly, and link to two freely available e-books where you can look up all the details for planning your system:
Watershed management. This means the enhancement of natural recharge into the aquifer and involves a modification of large parts of the watershed of a local river. You can look into permaculture design techniques (swales, keyline ploughing and ponds) for suitable techniques. This is often the most suitable technique because it also helps in water management on local farms and prevents soil erosion and other damage by runoff water.
Trees can play an important role in this technique. Up to a certain tree density of somewhere between 10% and 30% canopy cover, the benefits of trees for groundwater recharge outweigh the drawback that trees also consume more water. That is, when planting trees not too densely, they help to increase groundwater levels by keeping the pores in the soil open through which water can percolate down to the groundwater level. This is the result of some relatively new research, so the exact impacts and implications of this mechanism are not yet explored. The existing research work is available here and here, with an introductory article available as well.
Surface spreading. This category includes techniques such as recharge basins, modified stream beds, pits, shafts and others. The idea is to enhance percolation into the ground from surface area that catch stormwater. Maintenance often includes raking the bottom of recharge basins to prevent clogging from clay particles and other fines. Plus, this can only recharge into aquifers with no waterproof layers above them, and requires large land areas. But where this is possible, it is often the cheapest option.
Vadose zone wells. Also called recharge wells. Dry borehole wells or specifically drilled wells are used to recharge the aquifer. This can work with aquifers that have waterproof layers above them, but the main problem is clogging by fine particles over time, which cannot be effectively reversed. Overall, quite an expensive technique.
Aquifer storage and recovery (ASR) wells. Borehole wells that are designed both for recharge of the aquifer and recovery of water from the same well.
Relevant open access literature for further details and instructions:
Prospects for Managed Underground Storage of Recoverable Water. Published by National Research Council. 2008. 351 pages.
Source Book of Alternative Technologies for Freshwater Augmentation in Latin America and the Caribbean. Published by Unit of Sustainable Development and Environment, General Secretariat, Organization of American States. 1997. Here relevant: chapter "1.9 Artificial recharge of aquifers".
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