Yes, you can do research in AI with low funds (but you need a lot of time!). Notably because AI is not the same as applied machine learning (indeed running ML programs on big data requires a lot of computer power). For example, knowledge representation and reasoning or natural language processing (both are field of AI) generally don't require a lot of computer power.
A lot of recent papers (probably most of them) in journals like Artificial Intelligence, or in conferences like IJCAI (see their past proceedings) are somehow theoretical, and when something is implemented, it runs on a laptop or desktop. Notice that both AI journal and IJCAI conferences are peer-reviewed with a very selective process.
Actually it is difficult and rare to find a research paper in AI mentioning that costly equipment was needed to do the research. Costly supercomputers used for research are generally not used by AI researchers (but by researchers in physics or bioinformatics), and AI researchers often don't even have access to such facilities.
For examples, recent IJCAI2016 papers such as Coco: Runtime Reasoning About Conflicting Commitments, Interdependent Scheduling Games, Control of Fair Division, Verifying Pushdown Multi-Agent Systems against Strategy Logics, etc.... don't mention any costly computation. Actually, it is likely that most recent papers don't use and don't need large scale costly cloud computing. And some of that research might not have been implemented (perhaps by some intern) in anything more than a toy prototype.
Please include in your answer examples of recent research results and the required infrastructure that was used to create them.
Event recent IJCAI experimental papers like A Multicore Tool for Constraint Solving, Compiling Constraint Networks into Multivalued Decomposable Decision Graphs, External Memory Bidirectional Search, Multiple Constraint Acquisition, Completion of Disjunctive Logic Programs, Eliminating Disjunctions in Answer Set Programming by Restricted Unfolding and On the Empirical Time Complexity of Random 3-SAT at the Phase Transition mention at most a multi-core workstation (e.g. at most a dual Xeon socket workstation, often a cheaper laptop or desktop). I guess that most authors need their GPGU only as a graphics card, to use their display screen. BTW papers in other journals, such as JAIR (like Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning), when they mention computer equipement, gives similar kind of machines (laptop or at most high-end desktop).
(actually I don't remember having read an AI paper mentionning costly computing equipement; and I believe the reason for that is that in the academic community access to supercomputers is nearly reserved to research in other domains: physics, bioinformatics, etc... For an AI researcher gaining such access is difficult and uncommon; BTW in H2020 European research grants computing cost above 15% of the labor cost needs to be dully justified, so is exceptional)
However, you'll better publish your software as free software or open source, and you need a lot of time (preferably full-time, or at least half-time) to do the research work, publish it, and follow outside progress in your area. BTW, contacting a nearby university could be helpful (you could attend some seminars, etc...)
You can find interesting blogs (e.g. the one of J.Pitrat) of AI researchers working on just a laptop or a desktop.