You need a genome that isn't too complex, which suggests something microbial. You also need a genome that is well-characterised : either fully- sequenced, or with lots of sequenced genes to allow you to design primers. And finally you need a readily accessible source of material for DNA extraction.
I would suggest either dried yeast, as a source of Saccharomyces cerevisiae DNA, or live yoghurt as a source of DNA from Lactobacilli. You could even try culturing something from yoghurt to get pure bacterial colonies: since these would be from a food source there should be minimal risk.
You might be tempted by one of the traditional plant sources of DNA (e.g. onions) , but it can be tricky to avoid copurifying polysaccharides which inhibit DNA-related enzymes.
Finally, if you do culture some bacteria, or even if you use dried yeast, you probably wouldn't need to actually prepare DNA at all. PCR direct from bacterial cells is widely used - the 95 C step will release sufficient DNA.
added later in response to comment by OP
I chose to follow up on Lactobacillus delbrueckii subsp. bulgaricus because I found this coming up over and over again with reference to yoghurt. If you were to test a range of yoghurts I think you could be fairly confident that this would be present.
There are 5 published genomes for different strains of the species Lactobacillus delbrueckii including subspecies bulgaricus (various strains) and lactis. How different are they? I decided to look at one gene, and lactate dehydrogenase seemed like a logical choice (it produces the lactic acid).
The complete nucleotide sequence for the lactate dehydrogenase gene from Lactobacillus delbrueckii subsp. bulgaricus ATCC 11842 is here. If you run a BLAST from that page limiting the search to Lactobacilli (taxid 1578) you do indeed find hits to the gene from several other genomes. Here is a representative selection from the sequence alignment to the least similar of the hits:
Query 99766 CCTTCCACGCTTCTGGAGGTACTGAATTCGCCCTGGCTCAACTGGGCAACGATGCCGGTA 99825
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Sbjct 90616 CCTTCCACGCTTCTGAAGGTACTGAATTCGCCCTGGCCCAACTGGGCAACGATGCCGGTA 90675
Query 99826 TGTACGGTGCCGTTAAGATGGTTCTGTAATTTTGAATGTAAAAAAGCACCAGGCTTTTAA 99885
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Sbjct 90676 TGTACGGTGCCGTTAAGATGGTTCTGTAATTTTGAATATAAAAAAGCACCAGACTTTTAA 90735
Query 99886 AGTCTGGTGCtttttttGTTTATCCTAAAGAGTCCAGGGTTGCCTTTATCGTCGCGGCTG 99945
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Sbjct 90736 AGTCTGGTGCTTATTTTGTTTATTCTAAAGAGTCCAGGGTTGCCTTCATCGTCGCGGCTG 90795
Query 99946 AGGCCCGCATCTTGGCCAATTCGCTGTCATTTAACGGCAATTCCAGTACGTGGCTGATCC 100005
||||| |||||||||||||||||||||||||||||||||||||||| ||||||||||| |
Sbjct 90796 AGGCCTGCATCTTGGCCAATTCGCTGTCATTTAACGGCAATTCCAGCACGTGGCTGATTC 90855
Query 100006 CTTGTCCGTTGATGATGGCCAGGGTGCCCAGGTAGATCTCATCCTTGATCCCGTATTCCC 100065
|||| |||||||||||||||||||||||||| ||||||||||||||||||||||||||||
Sbjct 90856 CTTGCCCGTTGATGATGGCCAGGGTGCCCAGATAGATCTCATCCTTGATCCCGTATTCCC 90915
Query 100066 CGTGCAATGGTGCGGAAAGAGGCAGGGCCAGGTCGTTGTTTTCCAAAATGGCCGTCACGA 100125
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Sbjct 90916 CGTGCAATGGCGCGGAAAGAGGCAGGGCCAGGTCGTTGTTTTCCAAAATGGCCGCCACGA 90975
Query 100126 TCTTGGCCAGCATCATGGCCACGCCATAGTAGGTTGCGCCCTTTTTGACGATAATCTCGC 100185
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Sbjct 90976 TCTTGGCCAGCATCATGGCCACGCCATAGTAGGTCGCGCCCTTTTTGCCGATGATCTCGC 91035
As you can see, there are some differences, but there should be no problem designing primers matching both (again - this is just one part of the gene). Obviously you might wish to make an alignment for all 5 sequences, but I got the impression that at least some of the others were identical to the query, even though they are from different subspecies. These are very closely-related organisms.
In terms of your experiment, I suggest that you plan to use a range of yoghurts and/or starters as the source of template. You could even mix them, if there are constraints on your set-up.