The transactions of the society are available here online from 1965 to the present (apart from two that are missing.)
All paper copies of the transactions, from 1886/89 are available in the University of Leicester library.
Hint on searching: There are a lot of volumes here and it can be difficult to find what you are looking for. Suppose you’re looking for an article that contained the phrase (or was called): old wine in new bottles. You can type the following into Google (or your favourite search engine!) to find the issue you are looking for:
site:https://www.leicesterlitandphil.org.uk/ “old wine in new bottles”
(Don’t forget the space between … uk/ and “old wine … and the double quotation (“….”) marks around the phrase that you’re looking for.) This should then find the volume that you’re looking for.
Volume 59 – 1965
Volume 60 – 1966
Volume 61 – 1967 (missing)
Volume 62 – 1968
Volume 63 – 1969
Volume 64 – 1970
Volume 65 – 1971
Volume 66 – 1972
Volume 67 – 1973
Volume 68 – 1974
Volume 69 – 1975
Volume 70 – 1976
Volume 71 – 1977
Volume 72 – 1978
Volume 73 – 1979
Volume 74 – 1980
Volume 75 – 1981
Volume 76 – 1982
Volume 77 – 1983
—map to accompany volume 77
Volume 78 – 1984
Volume 79 – 1985
Volume 80 – 1986
Volume 81 – 1987
Volume 82 – 1988
Volume 83 – 1989
Volume 84 – 1990
Volume 85 – 1991
Volume 86 – 1992
Volume 87 – 1993
Volume 88 – 1994
Volume 89 – 1995
Volume 90 – 1996
Volume 91 – 1997
Volume 92 – 1998
Volume 93 – 1999
Volume 94 – 2000
Volume 95 – 2001
Volume 96 – 2002
Volume 97 – 2003
Volume 98 – 2004
Volume 99 – 2005
Volume 100 – 2006
Volume 101 – 2007
Volume 102 – 2008
Volume 103 – 2009
Volume 104 – 2010
Volume 105 – 2011
Volume 106 – 2012
Volume 107 – 2013
Volume 108 – 2014
Volume 109 – 2015
Volume 110 – 2016
Volume 111 – 2017
Volume 112 – 2018
Volume 113 – 2019
Volume 114 – 2020
Volume 115 – 2021